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Search: seq:1,1,2,3,5,8,13,21
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%I A000045 M0692 N0256
%S A000045 0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,
%T A000045 10946,17711,28657,46368,75025,121393,196418,317811,514229,832040,
%U A000045 1346269,2178309,3524578,5702887,9227465,14930352,24157817,39088169,63245986,102334155
%N A000045 Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.
%C A000045 Also sometimes called Lamé's sequence.
%C A000045 F(n+2) = number of binary sequences of length n that have no consecutive 0's.
%C A000045 F(n+2) = number of subsets of {1,2,...,n} that contain no consecutive integers.
%C A000045 F(n+1) = number of tilings of a 2 X n rectangle by 2 X 1 dominoes.
%C A000045 F(n+1) = number of matchings (i.e., Hosoya index) in a path graph on n vertices: F(5)=5 because the matchings of the path graph on the vertices A, B, C, D are the empty set, {AB}, {BC}, {CD} and {AB, CD}. - _Emeric Deutsch_, Jun 18 2001
%C A000045 F(n) = number of compositions of n+1 with no part equal to 1. [Cayley, Grimaldi]
%C A000045 Positive terms are the solutions to z = 2*x*y^4 + (x^2)*y^3 - 2*(x^3)*y^2 - y^5 - (x^4)*y + 2*y for x,y >= 0 (Ribenboim, page 193). When x=F(n), y=F(n + 1) and z > 0 then z=F(n + 1).
%C A000045 For Fibonacci search see Knuth, Vol. 3; Horowitz and Sahni; etc.
%C A000045 F(n) is the diagonal sum of the entries in Pascal's triangle at 45 degrees slope. - _Amarnath Murthy_, Dec 29 2001
%C A000045 F(n+1) is the number of perfect matchings in ladder graph L_n = P_2 X P_n. - Sharon Sela (sharonsela(AT)hotmail.com), May 19 2002
%C A000045 F(n+1) = number of (3412,132)-, (3412,213)- and (3412,321)-avoiding involutions in S_n.
%C A000045 This is also the Horadam sequence (0,1,1,1). - _Ross La Haye_, Aug 18 2003
%C A000045 An INVERT transform of A019590. INVERT([1,1,2,3,5,8,...]) gives A000129. INVERT([1,2,3,5,8,13,21,...]) gives A028859. - _Antti Karttunen_, Dec 12 2003
%C A000045 Number of meaningful differential operations of the k-th order on the space R^3. - _Branko Malesevic_, Mar 02 2004
%C A000045 F(n)=number of compositions of n-1 with no part greater than 2. Example: F(4)=3 because we have 3 = 1+1+1 = 1+2 = 2+1.
%C A000045 F(n) = number of compositions of n into odd parts; e.g., F(6) counts 1+1+1+1+1+1, 1+1+1+3, 1+1+3+1, 1+3+1+1, 1+5, 3+1+1+1, 3+3, 5+1. - _Clark Kimberling_, Jun 22 2004
%C A000045 F(n) = number of binary words of length n beginning with 0 and having all runlengths odd; e.g., F(6) counts 010101, 010111, 010001, 011101, 011111, 000101, 000111, 000001. - _Clark Kimberling_, Jun 22 2004
%C A000045 The number of sequences (s(0),s(1),...,s(n)) such that 0<s(i)<5, |s(i)-s(i-1)|=1 and s(0)=1 is F(n+1); e.g., F(5+1) = 8 corresponds to 121212, 121232, 121234, 123212, 123232, 123234, 123432, 123434. - _Clark Kimberling_, Jun 22 2004 [corrected by Neven Juric, Jan 09 2009]
%C A000045 Likewise F(6+1) = 13 corresponds to these thirteen sequences with seven numbers: 1212121, 1212123, 1212321, 1212323, 1212343, 1232121, 1232123, 1232321, 1232323, 1232343, 1234321, 1234323, 1234343. - Neven Juric, Jan 09 2008
%C A000045 A relationship between F(n) and the Mandelbrot set is discussed in the link "Le nombre d'or dans l'ensemble de Mandelbrot" (in French). - _Gerald McGarvey_, Sep 19 2004
%C A000045 For n>0, the continued fraction for F(2n-1)*Phi = [F(2n);L(2n-1),L(2n-1),L(2n-1),...] and the continued fraction for F(2n)*Phi=[F(2n+1)-1;1,L(2n)-2,1,L(2n)-2,...]. Also true: F(2n)*Phi=[F(2n+1);-L(2n),L(2n),-L(2n),L(2n),...] where L(i) is the i-th Lucas number (A000204).... - _Clark Kimberling_, Nov 28 2004 [corrected by _Hieronymus Fischer_, Oct 20 2010]
%C A000045 F(n+1) (for n >= 1) = number of permutations p of 1,2,3,...,n such that |k-p(k)| <= 1 for k=1,2,...,n. (For <= 2 and <= 3, see A002524 and A002526.) - _Clark Kimberling_, Nov 28 2004
%C A000045 The ratios F(n+1)/F(n) for n > 0 are the convergents to the simple continued fraction expansion of the golden section. - _Jonathan Sondow_, Dec 19 2004
%C A000045 Lengths of successive words (starting with a) under the substitution: {a -> ab, b -> a}. - _Jeroen F.J. Laros_, Jan 22 2005
%C A000045 The Fibonacci sequence, like any additive sequence, naturally tends to be geometric with common ratio not a rational power of 10; consequently, for a sufficiently large number of terms, Benford's law of first significant digit (i.e., first digit 1 <= d <= 9 occurring with probability log_10(d+1) - log_10(d)) holds. - _Lekraj Beedassy_, Apr 29 2005 (See Brown-Duncan, 1970. - _N. J. A. Sloane_, Feb 12 2017)
%C A000045 a(n) = Sum_{k=0..n} abs(A108299(n, k)). - _Reinhard Zumkeller_, Jun 01 2005
%C A000045 a(n) = A001222(A000304(n)).
%C A000045 F(n+2) = Sum_{k=0..n} binomial(floor((n+k)/2),k), row sums of A046854. - _Paul Barry_, Mar 11 2003
%C A000045 Number of order ideals of the "zig-zag" poset. See vol. 1, ch. 3, prob. 23 of Stanley. - _Mitch Harris_, Dec 27 2005
%C A000045 F(n+1)/F(n) is also the Farey fraction sequence (see A097545 for explanation) for the golden ratio, which is the only number whose Farey fractions and continued fractions are the same. - _Joshua Zucker_, May 08 2006
%C A000045 a(n+2) is the number of paths through 2 plates of glass with n reflections (reflections occurring at plate/plate or plate/air interfaces). Cf. A006356-A006359. - _Mitch Harris_, Jul 06 2006
%C A000045 F(n+1) equals the number of downsets (i.e., decreasing subsets) of an n-element fence, i.e., an ordered set of height 1 on {1,2,...,n} with 1 > 2 < 3 > 4 < ... n and no other comparabilities. Alternatively, F(n+1) equals the number of subsets A of {1,2,...,n} with the property that, if an odd k is in A, then the adjacent elements of {1,2,...,n} belong to A, i.e., both k - 1 and k + 1 are in A (provided they are in {1,2,...,n}). - _Brian Davey_, Aug 25 2006
%C A000045 Number of Kekulé structures in polyphenanthrenes. See the paper by Lukovits and Janezic for details. - _Parthasarathy Nambi_, Aug 22 2006
%C A000045 Inverse: With phi = (sqrt(5) + 1)/2, round(log_phi(sqrt((sqrt(5) a(n) + sqrt(5 a(n)^2 - 4))(sqrt(5) a(n) + sqrt(5 a(n)^2 + 4)))/2)) = n for n >= 3, obtained by rounding the arithmetic mean of the inverses given in A001519 and A001906. - David W. Cantrell (DWCantrell(AT)sigmaxi.net), Feb 19 2007
%C A000045 A result of Jacobi from 1848 states that every symmetric matrix over a p.i.d. is congruent to a triple-diagonal matrix. Consider the maximal number T(n) of summands in the determinant of an n X n triple-diagonal matrix. This is the same as the number of summands in such a determinant in which the main-, sub- and super-diagonal elements are all nonzero. By expanding on the first row we see that the sequence of T(n)'s is the Fibonacci sequence without the initial stammer on the 1's. - Larry Gerstein (gerstein(AT)math.ucsb.edu), Mar 30 2007
%C A000045 Suppose psi=log(phi). We get the representation F(n)=(2/sqrt(5))*sinh(n*psi) if n is even; F(n)=(2/sqrt(5))*cosh(n*psi) if n is odd. There is a similar representation for Lucas numbers (A000032). Many Fibonacci formulas now easily follow from appropriate sinh and cosh formulas. For example: the de Moivre theorem (cosh(x)+sinh(x))^m = cosh(mx)+sinh(mx) produces L(n)^2 + 5F(n)^2 = 2L(2n) and L(n)F(n) = F(2n) (setting x=n*psi and m=2). - _Hieronymus Fischer_, Apr 18 2007
%C A000045 Inverse: floor(log_phi(sqrt(5)*F(n)) + 1/2) = n, for n > 1. Also for n > 0, floor((1/2)*log_phi(5*F(n)*F(n+1))) = n. Extension valid for integer n, except n=0,-1: floor((1/2)*sign(F(n)*F(n+1))*log_phi|5*F(n)*F(n+1)|) = n (where sign(x) = sign of x). - _Hieronymus Fischer_, May 02 2007
%C A000045 F(n+2) = The number of Khalimsky-continuous functions with a two-point codomain. - Shiva Samieinia (shiva(AT)math.su.se), Oct 04 2007
%C A000045 From Kauffman and Lopes, Proposition 8.2, p. 21: "The sequence of the determinants of the Fibonacci sequence of rational knots is the Fibonacci sequence (of numbers)." - _Jonathan Vos Post_, Oct 26 2007
%C A000045 This is a_1(n) in the Doroslovacki reference.
%C A000045 Let phi = (sqrt(5)+1)/2 = 1.6180339...; then phi^n = (1/phi)*a(n) + a(n+1). Example: phi^4 = 6.8541019... = (0.6180339...)*3 + 5. Also phi = 1/1 + 1/2 + 1/(2*5) + 1/(5*13) + 1/(13*34) + 1/(34*89) + ... - _Gary W. Adamson_, Dec 15 2007
%C A000045 The sequence of first differences, F(n+1)-F(n), is essentially the same sequence: 1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... - _Colm Mulcahy_, Mar 03 2008
%C A000045 a(n) = the number of different ways to run up a staircase with n steps, taking steps of odd sizes where the order is relevant and there is no other restriction on the number or the size of each step taken. - _Mohammad K. Azarian_, May 21 2008
%C A000045 Equals row sums of triangle A144152. - _Gary W. Adamson_, Sep 12 2008
%C A000045 Except for the initial term, the numerator of the convergents to the recursion x = 1/(x+1). - _Cino Hilliard_, Sep 15 2008
%C A000045 F(n) is the number of possible binary sequences of length n that obey the sequential construction rule: if last symbol is 0, add the complement (1); else add 0 or 1. Here 0,1 are metasymbols for any 2-valued symbol set. This rule has obvious similarities to JFJ Laros's rule, but is based on addition rather than substitution and creates a tree rather than a single sequence. - _Ross Drewe_, Oct 05 2008
%C A000045 F(n) = Product_{k=1..(n-1)/2} (1 + 4*cos^2 k*Pi/n), where terms = roots to the Fibonacci product polynomials, A152063. - _Gary W. Adamson_, Nov 22 2008
%C A000045 Fp == 5^((p-1)/2) mod p, p = prime [Schroeder, p. 90]. - _Gary W. Adamson_ & _Alexander R. Povolotsky_, Feb 21 2009
%C A000045 (Ln)^2 - 5*(Fn)^2 = 4*(-1)^n. Example: 11^2 - 5*5 = -4. - _Gary W. Adamson_, Mar 11 2009
%C A000045 Output of Kasteleyn's formula for the number of perfect matchings of an m X n grid specializes to the Fibonacci sequence for m=2. - Sarah-Marie Belcastro (smbelcas(AT)toroidalsnark.net), Jul 04 2009
%C A000045 (F(n),F(n+4)) satisfies the Diophantine equation: X^2 + Y^2 - 7XY = 9*(-1)^n. - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 06 2009
%C A000045 (F(n),F(n+2)) satisfies the Diophantine equation: X^2 + Y^2 - 3XY = (-1)^n. - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 08 2009
%C A000045 a(n+2) = A083662(A131577(n)). - _Reinhard Zumkeller_, Sep 26 2009
%C A000045 Difference between number of closed walks of length n+1 from a node on a pentagon and number of walks of length n+1 between two adjacent nodes on a pentagon. - _Henry Bottomley_, Feb 10 2010
%C A000045 F(n+1) = number of Motzkin paths of length n having exactly one weak ascent. A Motzkin path of length n is a lattice path from (0,0) to (n,0) consisting of U=(1,1), D=(1,-1) and H=(1,0) steps and never going below the x-axis. A weak ascent in a Motzkin path is a maximal sequence of consecutive U and H steps. Example: a(5)=5 because we have (HHHH), (HHU)D, (HUH)D, (UHH)D, and (UU)DD (the unique weak ascent is shown between parentheses; see A114690). - _Emeric Deutsch_, Mar 11 2010
%C A000045 (F(n-1) + F(n+1))^2 - 5F(n-2)*F(n+2) = 9*(-1)^n. - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Mar 31 2010
%C A000045 From the Pinter and Ziegler reference's abstract: authors "show that essentially the Fibonacci sequence is the unique binary recurrence which contains infinitely many three-term arithmetic progressions. A criterion for general linear recurrences having infinitely many three-term arithmetic progressions is also given." - _Jonathan Vos Post_, May 22 2010
%C A000045 F(n+1) = number of paths of length n starting at initial node on the path graph P_4. - _Johannes W. Meijer_, May 27 2010
%C A000045 F(k) = Number of cyclotomic polynomials in denominator of generating function for number of ways to place k nonattacking queens on an n X n board. - _Vaclav Kotesovec_, Jun 07 2010
%C A000045 As n-> inf., (a(n)/a(n-1) - a(n-1)/a(n)) tends to 1.0. Example: a(12)/a(11) - a(11)/a(12) = 144/89 - 89/144 = 0.99992197.... - _Gary W. Adamson_, Jul 16 2010
%C A000045 From _Hieronymus Fischer_, Oct 20 2010: (Start)
%C A000045 Fibonacci numbers are those numbers m such that m*phi is closer to an integer than k*phi for all k, 1<=k<m. More formally: a(0)=0, a(1)=1, a(2)=1, a(n+1)=minimal m>a(n) such that m*phi is closer to an integer than a(n)*phi.
%C A000045 For all numbers 1 <= k < F(n), the inequality |k*phi-round(k*phi)| > |F(n)*phi-round(F(n)*phi)| holds.
%C A000045 F(n)*phi - round(F(n)*phi) = -((-phi)^(-n)), for n > 1.
%C A000045 Fract(1/2 + F(n)*phi) = 1/2 -(-phi)^(-n), for n > 1.
%C A000045 Fract(F(n)*phi) = (1/2)*(1 + (-1)^n) - (-phi)^(-n), n > 1.
%C A000045 Inverse: n = -log_phi |1/2 - fract(1/2 + F(n)*phi)|.
%C A000045 (End)
%C A000045 F(A001177(n)*k) mod n = 0, for any integer k. - _Gary Detlefs_, Nov 27 2010
%C A000045 F(n+k)^2 - F(n)^2 = F(k)*F(2n+k), for even k. - _Gary Detlefs_, Dec 04 2010
%C A000045 F(n+k)^2 + F(n)^2 = F(k)*F(2n+k), for odd k. - _Gary Detlefs_, Dec 04 2010
%C A000045 F(n) = round(phi* F(n-1)) for n > 1. - _Joseph P. Shoulak_, Jan 13 2012
%C A000045 For n > 0: a(n) = length of n-th row in Wythoff array A003603. - _Reinhard Zumkeller_, Jan 26 2012
%C A000045 From _Bridget Tenner_, Feb 22 2012: (Start)
%C A000045 The number of free permutations of [n].
%C A000045 The number of permutations of [n] for which s_k in supp(w) implies s_{k+-1} not in supp(w).
%C A000045 The number of permutations of [n] in which every decomposition into length(w) reflections is actually composed of simple reflections. (End)
%C A000045 The sequence F(n+1)^(1/n) is increasing. The sequence F(n+2)^(1/n) is decreasing. - _Thomas Ordowski_, Apr 19 2012
%C A000045 Two conjectures: For n > 1, F(n+2)^2 mod F(n+1)^2 = F(n)*F(n+1) - (-1)^n. For n > 0, (F(2n) + F(2n+2))^2 = F(4n+3) + Sum_{k = 2..2n} F(2k). - _Alex Ratushnyak_, May 06 2012
%C A000045 From _Ravi Kumar Davala_, Jan 30 2014: (Start)
%C A000045 Proof of Ratushnyak's first conjecture: For n > 1, F(n+2)^2 - F(n)*F(n+1) + (-1)^n = 2F(n+1)^2.
%C A000045 Consider: F(n+2)^2 - F(n)*F(n+1) - 2F(n+1)^2
%C A000045          = F(n+2)^2 - F(n+1)^2 - F(n+1)^2 - F(n)*F(n+1)
%C A000045          = (F(n+2) + F(n+1))*(F(n+2) - F(n+1)) - F(n+1)*(F(n+1) + F(n))
%C A000045          = F(n+3)*F(n) - F(n+1)*F(n+2) = -(-1)^n.
%C A000045 Proof of second conjecture: L(n) stands for Lucas number sequence from A000032.
%C A000045 Consider the fact that
%C A000045     L(2n+1)^2 = L(4n+2) - 2
%C A000045    (F(2n) + F(2n+2))^2 = F(4n+1) + F(4n+3) - 2
%C A000045    (F(2n) + F(2n+2))^2 = (Sum_{k = 2..2n} F(2k)) + F(4n+3).
%C A000045 (End)
%C A000045 The relationship: INVERT transform of (1,1,0,0,0,...) = (1, 2, 3, 5, 8, ...), while the INVERT transform of (1,0,1,0,1,0,1,...) = (1, 1, 2, 3, 5, 8, ...) is equivalent to: The numbers of compositions using parts 1 and 2 is equivalent to the numbers of compositions using parts == 1 mod 2 (i.e., the odd integers). Generally, the numbers of compositions using parts 1 and k is equivalent to the numbers of compositions of (n+1) using parts 1 mod k. Cf. A000930 for k = 3 and A003269 for k = 4. Example: for k = 2, n = 4 we have the compositions (22; 211, 121; 112; 1111) = 5; but using parts 1 and 3 we have for n = 5: (311, 131, 113, 11111, 5) = 5. - _Gary W. Adamson_, Jul 05 2012
%C A000045 The sequence F(n) is the binomial transformation of the alternating sequence (-1)^(n-1)*F(n), whereas the sequence F(n+1) is the binomial transformation of the alternating sequence (-1)^n*F(n-1). Both of these facts follow easily from the equalities a(n;1)=F(n+1) and b(n;1)=F(n) where a(n;d) and b(n;d) are so-called "delta-Fibonacci" numbers as defined in comments to A014445 (see also the papers of Witula et al.). - _Roman Witula_, Jul 24 2012
%C A000045 F(n) is the number of different (n-1)-digit binary numbers such that all substrings of length > 1 have at least one digit equal to 1. Example: for n = 5 there are 8 binary numbers with n - 1 = 4 digits (1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111), only the F(n) = 5 numbers 1010, 1011, 1101, 1110 and 1111 have the desired property. - _Hieronymus Fischer_, Nov 30 2012
%C A000045 For positive n, F(n+1) equals the determinant of the n X n tridiagonal matrix with 1's along the main diagonal, i's along the superdiagonal and along the subdiagonal where i = sqrt(-1). Example: Det([1,i,0,0; i,1,i,0; 0,i,1,i; 0,0,i,1]) = F(4+1) = 5. - _Philippe Deléham_, Feb 24 2013
%C A000045 For n >= 1, number of compositions of n where there is a drop between every second pair of parts, starting with the first and second part; see example. Also, a(n+1) is the number of compositions where there is a drop between every second pair of parts, starting with the second and third part; see example. - _Joerg Arndt_, May 21 2013
%C A000045 Central terms of triangles in A162741 and A208245, n > 0. - _Reinhard Zumkeller_, Jul 28 2013
%C A000045 For n >= 4, F(n-1) is the number of simple permutations in the geometric grid class given in A226433. - _Jay Pantone_, Sep 08 2013
%C A000045 a(n) are the pentagon (not pentagonal) numbers because the algebraic degree 2 number rho(5) = 2*cos(Pi/5) = phi (golden section), the length ratio diagonal/side in a pentagon, has minimal polynomial C(5,x) = x^2 - x - 1 (see A187360, n=5), hence rho(5)^n = a(n-1)*1 + a(n)*rho(5), n >= 0, in the power basis of the algebraic number field Q(rho(5)). One needs a(-1) = 1 here. See also the P. Steinbach reference under A049310. - _Wolfdieter Lang_, Oct 01 2013
%C A000045 A010056(a(n)) = 1. - _Reinhard Zumkeller_, Oct 10 2013
%C A000045 Define F(-n) to be F(n) for n odd and -F(n) for n even. Then for all n and k, F(n+2k)^2 - F(n)^2 = F(n+k)*( F(n+3k) - F(n-k) ). - _Charlie Marion_, Dec 20 2013
%C A000045 ( F(n), F(n+2k) ) satisfies the Diophantine equation: X^2 + Y^2 - L(2k)*X*Y = F(4k)^2*(-1)^n. This generalizes Bouhamida's comments dated Sep 06 2009 and Sep 08 2009. - _Charlie Marion_, Jan 07 2014
%C A000045 For any prime p there is an infinite periodic subsequence within F(n) divisible by p, that begins at index n = 0 with value 0, and its first nonzero term at n = A001602(i), and period k = A001602(i). Also see A236479. - _Richard R. Forberg_, Jan 26 2014
%C A000045 Range of row n of the circular Pascal array of order 5. - _Shaun V. Ault_, May 30 2014 [orig. Kicey-Klimko 2011, and observations by Glen Whitehead; more general work found in Ault-Kicey 2014]
%C A000045 Nonnegative range of the quintic polynomial 2*y - y^5 + 2*x*y^4 + x^2*y^3 - 2*x^3*y^2 - x^4*y with x, y >= 0, see Jones 1975. - _Charles R Greathouse IV_, Jun 01 2014
%C A000045 The expression round(1/(F(k+1)/F(n) + F(k)/F(n+1))), for n > 0, yields a Fibonacci sequence with k-1 leading zeros (with rounding 0.5 to 0). - _Richard R. Forberg_, Aug 04 2014
%C A000045 Conjecture: For n > 0, F(n) is the number of all admissible residue classes for which specific finite subsequences of the Collatz 3n + 1 function consists of n+2 terms. This has been verified for 0 < n < 51. For details see Links. - _Mike Winkler_, Oct 03 2014
%C A000045 a(4)=3 and a(6)=8 are the only Fibonacci numbers that are of the form prime+1. - _Emmanuel Vantieghem_, Oct 02 2014
%C A000045 a(1)=1=a(2), a(3)=2 are the only Fibonacci numbers that are of the form prime-1. - _Emmanuel Vantieghem_, Jun 07 2015
%C A000045 Any consecutive pair (m, k) of the Fibonacci sequence a(n) illustrates a fair equivalence between m miles and k kilometers. For instance, 8 miles ~ 13 km; 13 miles ~ 21 km. -_Lekraj Beedassy_, Oct 06 2014
%C A000045 Lim_{n -> oo} (log F(n+1)/log F(n))^n = e. - _Thomas Ordowski_, Oct 06 2014
%C A000045 a(n+1) counts closed walks on K_2, containing one loop on the other vertex. Equivalently the (1,1)_entry of A^(n+1) where the adjacency matrix of digraph is A=(0,1; 1,1). - _David Neil McGrath_, Oct 29 2014
%C A000045 a(n-1) counts closed walks on the graph G(1-vertex;l-loop,2-loop). - _David Neil McGrath_, Nov 26 2014
%C A000045 From _Tom Copeland_, Nov 02 2014: (Start)
%C A000045 Let P(x) = x/(1+x) with comp. inverse Pinv(x) = x/(1-x) = -P[-x], and C(x)= [1-sqrt(1-4x)]/2, an o.g.f. for the shifted Catalan numbers A000108, with inverse Cinv(x) = x * (1-x).
%C A000045 Fin(x) = P[C(x)] = C(x)/[1 + C(x)] is an o.g.f. for the Fine numbers, A000957 with inverse Fin^(-1)(x) = Cinv[Pinv(x)] = Cinv[-P(-x)].
%C A000045 Mot(x) = C[P(x)] = C[-Pinv(-x)] gives an o.g.f. for shifted A005043, the Motzkin or Riordan numbers with comp. inverse Mot^(-1)(x) = Pinv[Cinv(x)] = (x - x^2) / (1 - x + x^2) (cf. A057078).
%C A000045 BTC(x) = C[Pinv(x)] gives A007317, a binomial transform of the Catalan numbers, with BTC^(-1)(x) = P[Cinv(x)].
%C A000045 Fib(x) = -Fin[Cinv(Cinv(-x))] = -P[Cinv(-x)] = x + 2 x^2 + 3 x^3 + 5 x^4 + ... = (x+x^2)/[1-x-x^2] is an o.g.f. for the shifted Fibonacci sequence A000045, so the comp. inverse is Fib^(-1)(x) = -C[Pinv(-x)] = -BTC(-x) and Fib(x) = -BTC^(-1)(-x).
%C A000045 Generalizing to P(x,t) = x /(1 + t*x) and Pinv(x,t) = x /(1 - t*x) = -P(-x,t) gives other relations to lattice paths, such as the o.g.f. for A091867, C[P[x,1-t]], and that for A104597, Pinv[Cinv(x),t+1].
%C A000045 (End)
%C A000045 In keeping with historical accounts (see the references by P. Singh and S. Kak), the generalized Fibonacci sequence a, b, a + b, a + 2b, 2a + 3b, 3a + 5b, ... can also be described as the Gopala-Hemachandra numbers H(n) = H(n-1) + H(n-2), with F(n) = H(n) for a = b = 1, and Lucas sequence L(n) = H(n) for a = 2, b = 1. - _Lekraj Beedassy_, Jan 11 2015
%C A000045 D. E. Knuth writes: "Before Fibonacci wrote his work, the sequence F_{n} had already been discussed by Indian scholars, who had long been interested in rhythmic patterns that are formed from one-beat and two-beat notes. The number of such rhythms having n beats altogether is F_{n+1}; therefore both Gopāla (before 1135) and Hemachandra (c. 1150) mentioned the numbers 1, 2, 3, 5, 8, 13, 21, ... explicitly." (TAOCP Vol. 1, 2nd ed.) - _Peter Luschny_, Jan 11 2015
%C A000045 F(n+1) equals the number of binary words of length n avoiding runs of zeros of odd lengths. - _Milan Janjic_, Jan 28 2015
%C A000045 From _Russell Jay Hendel_, Apr 12 2015: (Start)
%C A000045 We prove Conjecture 1 of Rashid listed in the Formula section.
%C A000045 We use the following notation: F(n)=A000045(n), the Fibonacci numbers, and L(n) = A000032(n), the Lucas numbers. The fundamental Fibonacci-Lucas recursion asserts that G(n) = G(n-1)+ G(n-2), with "L" or "F" replacing "G".
%C A000045 We need the following prerequisites which we label (A), (B),(C), (D). The prerequisites are formulas in the Koshy book listed in the References section. (A) F(m-1)+F(m+1) = L(m) (Koshy, p. 97, #32), (B) L(2m)+2(-1)^m = L(m)^2 (Koshy p. 97, #41), (C) F(m+k)F(m-k) = (-1)^n F(k)^2 (Koshy, p. 113, #24, Tagiuri's identity), and (D) F(n)^2+F(n+1)^2 = F(2n+1) (Koshy, p. 97, #30).
%C A000045 We must also prove (E), L(n+2) F(n-1) = F(2n+1)+2(-1)^n. To prove (E), first note that by (A), proof of (E) is equivalent to proving that F(n+1)F(n-1) + F(n+3)F(n-1) = F(2n+1)+2(-1)^n. But by (C) with k=1, we have F(n+1)F(n-1) = F(n)^2 +(-1)^n. Applying (C) again with k=2 and m=n+1, we have F(n+3)F(n-1) = F(n+1)+(-1)^n. Adding these two applications of (C) together and using (D) we have, F(n+1)F(n-1) + F(n+3)F(n-1) = F(n)^2 + F(n+1)^2 + 2(-1)^n = F(2n+1)+2(-1)^n, completing the proof of (E).
%C A000045 We now prove Conjecture 1. By (A) and the Fibonacci-Lucas recursion, we have F(2n+1)+F(2n+2)+F(2n+3)+F(2n+4) = [F(2n+1)+F(2n+3)] + [F(2n+2)+F(2n+4)] = L(2n+2)+L(2n+3)=L(2n+4). But then by (B), with m=2n+4, we have sqrt(L(2n+4)+2(-1)^n)) = L(n+2). Finally by (E), we have L(n+2) F(n-1)= F(2n+1)+2*(-1)^n. Dividing both sides by F(n-1), we have (F(2n+1)+2*(-1)^n)/F(n-1) = L(n+2) = sqrt(F(2n+1)+F(2n+2)+F(2n+3)+F(2n+4)+2(-1)^n), as required.
%C A000045 (End)
%C A000045 In Fibonacci's Liber Abaci the rabbit problem appears in the translation of L. E. Sigler on pp. 404-405, and a remark [27] on p. 637. - _Wolfdieter Lang_, Apr 17 2015
%C A000045 a(n) counts partially ordered partitions of (n-1) into parts 1,2,3 where only the order of adjacent 1's and 2's are unimportant. (See example.) - _David Neil McGrath_, Jul 27 2015
%C A000045 F(n) divides F(nk). Proved by Marjorie Bicknell and Verner E Hoggatt Jr. - _Juhani Heino_, Aug 24 2015
%C A000045 F(n) is the number of UDU-equivalence classes of ballot paths of length n. Two ballot paths of length n with steps U = (1,1), D = (1,-1) are UDU-equivalent whenever the positions of UDU are the same in both paths. - _Kostas Manes_, Aug 25 2015
%C A000045 Cassini's identity F(2n+1) * F(2n+3) = F(2n+2)^2 + 1 is the basis for a geometrical paradox (or dissection fallacy) in A262342. - _Jonathan Sondow_, Oct 23 2015
%C A000045 For n >= 4, F(n) is the number of up-down words on alphabet {1,2,3} of length n-2. - _Ran Pan_, Nov 23 2015
%C A000045 F(n+2) is the number of terms in p(n), where p(n)/q(n) is the n-th convergent of the formal infinite continued fraction [a(0),a(1),...]; e.g., p(3) = a(0)a(1)a(2)a(3) + a(0)a(1) + a(0)a(3) + a(2)a(3) + 1 has F(5) terms. Also, F(n+1) is the number of terms in q(n). - _Clark Kimberling_, Dec 23 2015
%C A000045 F(n+1) (for n >= 1) is the permanent of an n X n matrix M with M(i,j)=1 if |i-j| <= 1 and 0 otherwise. - _Dmitry Efimov_, Jan 08 2016
%C A000045 A trapezoid has three sides of lengths in order F(n), F(n+2), F(n). For increasing n a very close approximation to the maximum area will have the fourth side equal to 2*F(n+1). For a trapezoid with lengths of sides in order F(n+2), F(n), F(n+2), the fourth side will be F(n+3). - _J. M. Bergot_, Mar 17 2016
%C A000045 (1) Join two triangles with lengths of sides L(n), F(n+3), L(n+2) and F(n+2), L(n+1), L(n+2) (where L(n)=A000032(n)) along the common side of length L(n+2) to create an irregular quadrilateral. Its area is approximately (5*F(2*n-1) - (F(2*n-7) - F(2*n-13))/5. (2) Join two triangles with lengths of sides L(n), F(n+2), F(n+3) and L(n+1), F(n+1, F(n+3) along the common side F(n+3) to form an irregular quadrilateral. Its area is approximately 4*F(2*n-1) - 2*(F(2*n-7) + F(2*n-18)). - _J. M. Bergot_, Apr 06 2016
%C A000045 From _Clark Kimberling_, Jun 13 2016: (Start)
%C A000045 Let T* be the infinite tree with root 0 generated by these rules: if p is in T*, then p+1 is in T* and x*p is in T*.
%C A000045 Let g(n) be the set of nodes in the n-th generation, so that g(0) = {0}, g(1) = {1}, g(2) = {2, x}, g(3) = {3, 2x, x+1, x^2}, etc.
%C A000045 Let T(r) be the tree obtained by substituting r for x.
%C A000045 If a positive integer N is not a square and r = sqrt(N), then the number of (not necessarily distinct) integers in g(n) is A000045(n), for n > = 1. See A274142. (End)
%C A000045 Consider the partitions of n, with all summands initially listed in nonincreasing order. Freeze all the 1's in place and then allow all the other summands to change their order, without displacing any of the 1's. The resulting number of arrangements is a(n+1). - _Gregory L. Simay_, Jun 14 2016
%C A000045 Limit of the matrix power M^k shown in A163733, Sep 14 2016; as k-->inf. results in a single column vector equal to the Fibonacci sequence. - _Gary W. Adamson_, Sep 19 2016
%C A000045 F(n) and Lucas numbers L(n), being related by the formulas F(n) = (F(n-1) + L(n-1))/2 and L(n) = 2 F(n+1) - F(n), are a typical pair of "autosequences" (see the link to OEIS Wiki). - _Jean-François Alcover_, Jun 10 2017
%C A000045 Also the number of independent vertex sets and vertex covers in the (n-2)-path graph. - _Eric W. Weisstein_, Sep 22 2017
%C A000045 Shifted numbers of {UD, DU, FD, DF}-equivalence classes of Łukasiewicz paths. Łukasiewicz paths are P-equivalent iff the positions of pattern P are identical in these paths. - _Sergey Kirgizov_, Apr 08 2018
%C A000045 For n>0, F(n) = the number of Markov equivalence classes with skeleton the path on n nodes. See Theorem 2.1 in the article by A. Radhakrishnan et al. below. - _Liam Solus_, Aug 23 2018
%C A000045 For n >= 2, also: number of terms in A032858 (every other base-3 digit is strictly smaller than its neighbors) with n-2 digits in base 3. - _M. F. Hasler_, Oct 05 2018
%C A000045 F(n+1) is the number of fixed points of the Foata transformation on S_n. - _Kevin Long_, Oct 17 2018
%C A000045 F(n+2) is the dimension of the Hecke algebra of type A_n with independent parameters (0,1,0,1,...) or (1,0,1,0,...). See Corollary 1.5 in the link "Hecke algebras with independent parameters". - _Jia Huang_, Jan 20 2019
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%H A000045 Mark A. Shattuck and Carl G. Wagner, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Shattuck/shattuck56.html">Periodicity and Parity Theorems for a Statistic on r-Mino Arrangements</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.3.6.
%H A000045 S. Silvia, <a href="http://arttech.about.com/library/weekly/aa060900a_fibonacci_sequence.htm">Fibonacci sequence</a> [broken link]
%H A000045 Parmanand Singh, <a href="http://dx.doi.org/10.1016/0315-0860(85)90021-7">The so-called Fibonacci numbers in ancient and medieval India</a>, Historia Mathematica, Volume 12 (3), 1985, 229-244.
%H A000045 Jaap Spies, <a href="http://www.jaapspies.nl/oeis/a000045.sage">Sage program for computing A000045</a>
%H A000045 Michael Z. Spivey and Laura L. Steil, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.html">The k-Binomial Transforms and the Hankel Transform</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
%H A000045 Z.-H. Sun, <a href="http://202.195.112.2/xsjl/szh/ConFn.pdf">Congruences For Fibonacci Numbers</a>
%H A000045 Asep K. Supriatna, Ema Carnia, Meksianis Z. Ndii, <a href="https://doi.org/10.1016/j.heliyon.2019.e01130">Fibonacci numbers: A population dynamics perspective</a>, Heliyon (2019) Vol. 5, Issue 1, e01130.
%H A000045 Roberto Tauraso, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Tauraso/tauraso3.html">A New Domino Tiling Sequence</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.3.
%H A000045 Thesaurus.Maths.org, <a href="http://thesaurus.maths.org/dictionary/map/word/3788">Fibonacci sequence</a> [broken link]
%H A000045 K. Tognetti, <a href="/A000045/a000045.pdf">Letter to N. J. A. Sloane (with attachments), May 25 1994</a>
%H A000045 K. Tognetti, <a href="/A000045/a000045_2.pdf">The Search for the Golden Sequence</a>, Draft Manuscript, May 25 1994.
%H A000045 K. Tognetti, <a href="http://www.austms.org.au/Modules/Fib">Fibonacci-His Rabbits and His Numbers and Kepler</a>
%H A000045 Tony van Ravenstein, <a href="/A000045/a000045_1.pdf">The three gap theorem (Steinhaus conjecture)</a>, Journal of the Australian Mathematical Society (Series A) 45.03 (1988): 360-370. [Annotated scanned copy]
%H A000045 C. Vila, <a href="http://www.boingboing.net/2010/03/22/dreamlike-animation.html">Nature by numbers</a> (animation).
%H A000045 Christobal Vila, <a href="http://wimp.com/naturenumbers/">Nature Numbers</a> (Video related to Fibonacci numbers)
%H A000045 N. N. Vorob'ev, <a href="http://eom.springer.de/F/f040020.htm">Fibonacci numbers</a>, Springer's Encyclopaedia of Mathematics.
%H A000045 A. Y. Z. Wang, P. Wen, <a href="https://doi.org/10.1186/s13660-015-0595-6">On the partial finite sums of the reciprocals of the Fibonacci numbers</a>, Journal of Inequalities and Applications, 2015.
%H A000045 Carl G. Wagner, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Wagner/wagner3.html">Partition Statistics and q-Bell Numbers (q = -1)</a>, J. Integer Seqs., Vol. 7, 2004.
%H A000045 Robert Walker, <a href="http://www.youtube.com/watch?v=Wx4ZfuMl-FI&amp;NR=1">Inharmonic "Golden Rhythmicon" - Fibonacci Sequence in Pairs Approaching Golden Ratio - With Bounce</a>
%H A000045 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a>, <a href="http://mathworld.wolfram.com/Double-FreeSet.html">Double-Free Set</a>, <a href="http://mathworld.wolfram.com/Fibonaccin-StepNumber.html">Fibonacci n-Step Number</a>, <a href="http://mathworld.wolfram.com/ResistorNetwork.html">Resistor Network</a>
%H A000045 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HosoyaIndex.html">Hosoya Index</a>
%H A000045 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IndependentVertexSet.html">Independent Vertex Set</a>
%H A000045 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Matching.html">Matching</a>
%H A000045 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PathGraph.html">Path Graph</a>
%H A000045 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/VertexCover.html">Vertex Cover</a>
%H A000045 Wikipedia, <a href="http://www.wikipedia.org/wiki/Fibonacci_number">Fibonacci number</a>
%H A000045 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cassini_and_Catalan_identities">Cassini and Catalan identities</a>
%H A000045 Willem's Fibonacci site, <a href="http://home.zonnet.nl/LeonardEuler/fiboe.htm">Fibonacci</a>
%H A000045 Mike Winkler, <a href="http://arxiv.org/abs/1412.0519">On the structure and the behaviour of Collatz 3n + 1 sequences - Finite subsequences and the role of the Fibonacci sequence</a>, arXiv:1412.0519 [math.GM], 2014.
%H A000045 Roman Witula, Damian Slota and Edyta Hetmaniok, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AMI_41_from255to263.pdf">Bridges between different known integer sequences</a>, Annales Mathematicae et Informaticae, 41 (2013) pp. 255-263.
%H A000045 R. Yanco, <a href="/A007380/a007380.pdf">Letter and Email to N. J. A. Sloane, 1994</a>
%H A000045 R. Yanco and A. Bagchi, <a href="/A007380/a007380_1.pdf">K-th order maximal independent sets in path and cycle graphs</a>, Unpublished manuscript, 1994. (Annotated scanned copy)
%H A000045 Donovan Young, <a href="https://www.emis.de/journals/JIS/VOL21/Young/young2.html">The Number of Domino Matchings in the Game of Memory</a>, J. Int. Seq., Vol. 21 (2018), Article 18.8.1.
%H A000045 Aimei Yu and Xuezheng Lv, <a href="http://dx.doi.org/10.1007/s10910-006-9088-7">The Merrifield-Simmons indices and Hosoya indices of trees with k pendant vertices</a>, J. Math. Chem., Vol. 41 (2007), pp. 33-43. See page 35.
%H A000045 Tianping Zhang and Yuankui Ma, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Zhang/zhang56.html">On Generalized Fibonacci Polynomials and Bernoulli Numbers</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.3.
%H A000045 <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H A000045 <a href="/index/Di#divseq">Index to divisibility sequences</a>
%H A000045 <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%H A000045 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1).
%H A000045 <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>
%H A000045 <a href="/index/Be#Benford">Index entries for sequences related to Benford's law</a>
%F A000045 G.f.: x / (1 - x - x^2).
%F A000045 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (k + x)/(1 + k*x). - _Paul D. Hanna_, Oct 26 2013
%F A000045 F(n) = ((1+sqrt(5))^n - (1-sqrt(5))^n)/(2^n*sqrt(5)).
%F A000045 Alternatively, F(n) = ((1/2+sqrt(5)/2)^n - (1/2-sqrt(5)/2)^n)/sqrt(5).
%F A000045 F(n) = F(n-1) + F(n-2) = -(-1)^n F(-n).
%F A000045 F(n) = round(phi^n/sqrt(5)).
%F A000045 F(n+1) = Sum_{j=0..floor(n/2)} binomial(n-j, j).
%F A000045 A strong divisibility sequence, that is, gcd(a(n), a(m)) = a(gcd(n, m)) for all positive integers n and m. - _Michael Somos_, Jan 03 2017
%F A000045 E.g.f.: (2/sqrt(5))*exp(x/2)*sinh(sqrt(5)*x/2). - _Len Smiley_, Nov 30 2001
%F A000045 [0 1; 1 1]^n [0 1] = [F(n); F(n+1)]
%F A000045 x | F(n) ==> x | F(kn).
%F A000045 A sufficient condition for F(m) to be divisible by a prime p is (p - 1) divides m, if p == 1 or 4 (mod 5); (p + 1) divides m, if p == 2 or 3 (mod 5); or 5 divides m, if p = 5. (This is essentially Theorem 180 in Hardy and Wright.) - Fred W. Helenius (fredh(AT)ix.netcom.com), Jun 29 2001
%F A000045 a(n)=F(n) has the property: F(n)*F(m) + F(n+1)*F(m+1) = F(n+m+1). - _Miklos Kristof_, Nov 13 2003
%F A000045 From _Kurmang. Aziz. Rashid_, Feb 21 2004: (Start)
%F A000045 Conjecture 1: for n >= 2, sqrt(F(2n+1) + F(2n+2) + F(2n+3) + F(2n+4) + 2*(-1)^n) = (F(2n+1) + 2*(-1)^n)/F(n-1). [For a proof see Comments section.]
%F A000045 Conjecture 2: for n >= 0, (F(n+2)*F(n+3)) - (F(n+1)*F(n+4)) + (-1)^n = 0.
%F A000045 [Two more conjectures removed by _Peter Luschny_, Nov 17 2017 ]
%F A000045 Theorem 1: for n >= 0, (F(n+3)^ 2 - F(n+1)^ 2)/F(n+2) = (F(n+3)+ F(n+1)).
%F A000045 Theorem 2: for n >= 0, F(n+10) = 11*F(n+5) + F(n).
%F A000045 Theorem 3: for n >= 6, F(n) = 4*F(n-3) + F(n-6). (End)
%F A000045 Conjecture 2 of Rashid is actually a special case of the general law F(n)*F(m) + F(n+1)*F(m+1) = F(n+m+1) (take n <- n+1 and m <- -(n+4) in this law). - Harmel Nestra (harmel.nestra(AT)ut.ee), Apr 22 2005
%F A000045 Conjecture 2 of Rashid Kurmang simplified: F(n)*F(n+3) = F(n+1)*F(n+2)-(-1)^n. Follows from d'Ocagne's identity: m=n+2. - _Alex Ratushnyak_, May 06 2012
%F A000045 Conjecture: for all c such that 2-Phi <= c < 2*(2-Phi) we have F(n) = floor(Phi*a(n-1)+c) for n > 2. - _Gerald McGarvey_, Jul 21 2004
%F A000045 |2*F(n) - 9*F(n+1)| = 4*A000032(n) + A000032(n+1). - _Creighton Dement_, Aug 13 2004
%F A000045 For x > Phi, Sum_{n>=0} F(n)/x^n = x/(x^2 - x - 1) - _Gerald McGarvey_, Oct 27 2004
%F A000045 F(n+1) = exponent of the n-th term in the series f(x, 1) determined by the equation f(x, y) = xy + f(xy, x). - _Jonathan Sondow_, Dec 19 2004
%F A000045 a(n-1) = Sum_{k=0..n} (-1)^k*binomial(n-ceiling(k/2), floor(k/2)). - _Benoit Cloitre_, May 05 2005
%F A000045 F(n+1) = Sum_{k=0..n} binomial((n+k)/2, (n-k)/2)(1+(-1)^(n-k))/2. - _Paul Barry_, Aug 28 2005
%F A000045 Fibonacci(n) = Product_{j=1..ceiling(n/2)-1} (1 + 4(cos(j*Pi/n))^2). [Bicknell and Hoggatt, pp. 47-48.] - _Emeric Deutsch_, Oct 15 2006
%F A000045 F(n) = 2^-(n-1)*Sum_{k=0..floor((n-1)/2)} binomial(n,2*k+1)*5^k. - _Hieronymus Fischer_, Feb 07 2006
%F A000045 a(n) = (b(n+1) + b(n-1))/n where {b(n)} is the sequence A001629. - _Sergio Falcon_, Nov 22 2006
%F A000045 F(n*m) = Sum_{k = 0..m} binomial(m,k)*F(n-1)^k*F(n)^(m-k)*F(m-k). The generating function of F(n*m) (n fixed, m = 0,1,2,...) is G(x) = F(n)*x / ((1 - F(n-1)*x)^2 - F(n)*x*(1 - F(n-1)*x) - (F(n)*x)^2). E.g., F(15) = 610 = F(5*3) = binomial(3,0)* F(4)^0*F(5)^3*F(3) + binomial(3,1)* F(4)^1*F(5)^2*F(2) + binomial(3,2)* F(4)^2*F(5)^1*F(1) + binomial(3,3)* F(4)^3*F(5)^0*F(0) = 1*1*125*2 + 3*3*25*1 + 3*9*5*1 + 1*27*1*0 = 250 + 225 + 135 + 0 = 610. - _Miklos Kristof_, Feb 12 2007
%F A000045 From _Miklos Kristof_, Mar 19 2007: (Start)
%F A000045   Let L(n) = A000032(n) = Lucas numbers. Then:
%F A000045   For a >= b and odd b,  F(a+b) + F(a-b) = L(a)*F(b).
%F A000045   For a >= b and even b, F(a+b) + F(a-b) = F(a)*L(b).
%F A000045   For a >= b and odd b,  F(a+b) - F(a-b) = F(a)*L(b).
%F A000045   For a >= b and even b, F(a+b) - F(a-b) = L(a)*F(b).
%F A000045   F(n+m) + (-1)^m*F(n-m) = F(n)*L(m);
%F A000045   F(n+m) - (-1)^m*F(n-m) = L(n)*F(m);
%F A000045   F(n+m+k) + (-1)^k*F(n+m-k) + (-1)^m*(F(n-m+k) + (-1)^k*F(n-m-k)) = F(n)*L(m)*L(k);
%F A000045   F(n+m+k) - (-1)^k*F(n+m-k) + (-1)^m*(F(n-m+k) - (-1)^k*F(n-m-k)) = L(n)*L(m)*F(k);
%F A000045   F(n+m+k) + (-1)^k*F(n+m-k) - (-1)^m*(F(n-m+k) + (-1)^k*F(n-m-k)) = L(n)*F(m)*L(k);
%F A000045   F(n+m+k) - (-1)^k*F(n+m-k) - (-1)^m*(F(n-m+k) - (-1)^k*F(n-m-k)) = 5*F(n)*F(m)*F(k). (End)
%F A000045 A corollary to Kristof 2007 is 2*F(a+b) = F(a)*L(b) + L(a)*F(b). - _Graeme McRae_, Apr 24 2014
%F A000045 For n > m, the sum of the 2m consecutive Fibonacci numbers F(n-m-1) thru F(n+m-2) is F(n)*L(m) if m is odd, and L(n)*F(m) if m is even (see the McRae link). - _Graeme McRae_, Apr 24 2014.
%F A000045 F(n) = b(n) + (p-1)*Sum_{k=2..n-1} floor(b(k)/p)*F(n-k+1) where b(k) is the digital sum analog of the Fibonacci recurrence, defined by b(k) = ds_p(b(k-1)) + ds_p(b(k-2)), b(0)=0, b(1)=1, ds_p=digital sum base p. Example for base p=10: F(n) = A010077(n) + 9*Sum_{k=2..n-1} A059995(A010077(k))*F(n-k+1). - _Hieronymus Fischer_, Jul 01 2007
%F A000045 F(n) = b(n)+p*Sum_{k=2..n-1} floor(b(k)/p)*F(n-k+1) where b(k) is the digital product analog of the Fonacci recurrence, defined by b(k) = dp_p(b(k-1)) + dp_p(b(k-2)), b(0)=0, b(1)=1, dp_p=digital product base p. Example for base p=10: F(n) = A074867(n) + 10*Sum_{k=2..n-1} A059995(A074867(k))*F(n-k+1). - _Hieronymus Fischer_, Jul 01 2007
%F A000045 a(n) = denominator of continued fraction [1,1,1,...] (with n ones); e.g., 2/3 = continued fraction [1,1,1]; where barover[1] = [1,1,1,...] = 0.6180339.... - _Gary W. Adamson_, Nov 29 2007
%F A000045 F(n + 3) = 2F(n + 2) - F(n), F(n + 4) = 3F(n + 2) - F(n), F(n + 8) = 7F(n + 4) - F(n), F(n + 12) = 18F(n + 6) - F(n). - _Paul Curtz_, Feb 01 2008
%F A000045 1 = 1/(1*2) + 1/(1*3) + 1/(2*5) + 1/(3*8) + 1/(5*13) + ... = 1/2 + 1/3 + 1/10 + 1/24 + 1/65 + 1/168 + ...; where A059929 = (0, 2, 3, 10, 24, 65, 168, ...). - _Gary W. Adamson_, Mar 16 2008
%F A000045 a(2^n) = Product_{i=0..n-2} B(i) where B(i) is A001566. Example 3*7*47 = F(16). - _Kenneth J Ramsey_, Apr 23 2008
%F A000045 F(n) = (1/(n-1)!) * (n^(n-1) - (C(n-2,0) + 4*C(n-2,1) + 3*C(n-2,2))*n^(n-2) + (10*C(n-3,0) + 49*C(n-3,1) + 95*C(n-3,2) + 83*C(n-3,3) + 27*C(n-3,4))*n^(n-3) - (90*C(n-4,0) + 740*C(n-4,1) + 2415*C(n-4,2) + 4110*C(n-4,3) + 3890*C(n-4,4) + 1950*C(n-4,5) + 405*C(n-4,6))*n^(n-4) + ... ). - _André F. Labossière_, Nov 24 2004
%F A000045 a(n+1) = Sum_{k=0..n} A109466(n,k)*(-1)^(n-k). -_Philippe Deléham_, Oct 26 2008
%F A000045 a(n) = Sum_{l_1=0..n+1} Sum_{l_2=0..n}...Sum_{l_i=0..n-i}... Sum_{l_n=0..1} delta(l_1,l_2,...,l_i,...,l_n), where delta(l_1,l_2,...,l_i,...,l_n) = 0 if any l_i + l_(i+1) >= 2 for i=1..n-1 and delta(l_1,l_2,...,l_i,...,l_n) = 1 otherwise. - _Thomas Wieder_, Feb 25 2009
%F A000045 a(n+1) = 2^n sqrt(Product_{k=1..n} cos(k Pi/(n+1))^2+1/4)) (Kasteleyn's formula specialized). - Sarah-Marie Belcastro (smbelcas(AT)toroidalsnark.net), Jul 04 2009
%F A000045 a(n+1) = Sum_{k=floor(n/2) mod 5} C(n,k) - Sum_{k=floor((n+5)/2) mod 5} C(n,k) = A173125(n) - A173126(n) = |A054877(n)-A052964(n-1)|. - _Henry Bottomley_, Feb 10 2010
%F A000045 If p[i] = modp(i,2) and if A is Hessenberg matrix of order n defined by: A[i,j] = p[j-i+1], (i <= j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n >= 1, a(n)=det A. - _Milan Janjic_, May 02 2010
%F A000045 Lim_{k->infinity} F(k+n)/F(k) = (L(n) + F(n)*sqrt(5))/2 with the Lucas numbers L(n)= A000032(n). - _Johannes W. Meijer_, May 27 2010
%F A000045 For n >= 1, F(n) = round(log_2(2^(phi*F(n-1)) + 2^(phi*F(n-2)))), where phi is the golden ratio. - _Vladimir Shevelev_, Jun 24 2010, Jun 27 2010
%F A000045 For n >= 1, a(n+1) = ceiling(phi*a(n)), if n is even and a(n+1) = floor(phi*a(n)), if n is odd (phi = golden ratio). - _Vladimir Shevelev_, Jul 01 2010
%F A000045 a(n) = 2*a(n-2) + a(n-3), n > 2. - _Gary Detlefs_, Sep 08 2010
%F A000045 a(2^n) = Product_{i=0..n-1} A000032(2^i). - _Vladimir Shevelev_, Nov 28 2010
%F A000045 a(n)^2 - a(n-1)^2 = a(n+1)*a(n-2), see A121646.
%F A000045 a(n) = sqrt((-1)^k*(a(n+k)^2 - a(k)*a(2n+k))), for any k. - _Gary Detlefs_, Dec 03 2010
%F A000045 F(2*n) = F(n+2)^2 - F(n+1)^2 - 2*F(n)^2. - _Richard R. Forberg_, Jun 04 2011
%F A000045 (-1)^(n+1) = F(n)^2 + F(n)*F(1+n) - F(1+n)^2.
%F A000045   F(n) = -F(n+2)(-2 + (F(n+1))^4 + 2*(F(n+1)^3*F(n+2)) - (F(n+1)*F(n+2))^2 2*F(n+1)(F(n+2))^3 + (F(n+2))^4)- F(n+1). - _Artur Jasinski_, Nov 17 2011
%F A000045 F(n) = 1 + Sum_{x=1..n-2} F(x). - _Joseph P. Shoulak_, Feb 05 2012
%F A000045 F(n) = 4*F(n-2) - 2*F(n-3) - F(n-6). - _Gary Detlefs_, Apr 01 2012
%F A000045 F(n) = round(phi^(n+1)/(phi+2)). - _Thomas Ordowski_, Apr 20 2012
%F A000045 From _Sergei N. Gladkovskii_, Jun 03 2012: (Start)
%F A000045 G.f. A(x) = x/(1-x-x^2) = G(0)/sqrt(5) where G(k)= 1 -((-1)^k)*2^k/(a^k - b*x*a^k*2^k/(b*x*2^k - 2*((-1)^k)*c^k/G(k+1))) and a=3+sqrt(5), b=1+sqrt(5), c=3-sqrt(5); (continued fraction, 3rd kind, 3-step).
%F A000045 Let E(x) be the e.g.f., i.e.,
%F A000045 E(x) = 1*x + 1/2*x^2 + 1/3*x^3 + 1/8*x^4 + 1/24*x^5 + 1/90*x^6 + 13/5040*x^7 + ...; then
%F A000045 E(x) = G(0)/sqrt(5); G(k)= 1 -((-1)^k)*2^k/(a^k - b*x*a^k*2^k/(b*x*2^k - 2*((-1)^k)*(k+1)*c^k/G(k+1))), where a=3+sqrt(5), b=1+sqrt(5), c=3-sqrt(5); (continued fraction, 3rd kind, 3-step).
%F A000045 (End)
%F A000045 From _Hieronymus Fischer_, Nov 30 2012: (Start)
%F A000045 F(n) = 1 + Sum_{j_1=1..n-2} 1 + Sum_{j_1=1..n-2} Sum_{j_2=1..j_1-2} 1 + Sum_{j_1=1..n-2} Sum_{j_2=1..j_1-2} Sum_{j_3=1..j_2-2} 1 + ... + Sum_{j_1=1..n-2} Sum_{j_2=1..j_1-2} Sum_{j_3=1..j_2-2} ... Sum_{j_k=1..j_(k-1)-2} 1, where k = floor((n-1)/2).
%F A000045 Example: F(6) = 1 + Sum_{j=1..4} 1 + Sum_{j=1..4} Sum_{k=1..(j-2)} 1 + 0 = 1 + (1 + 1 + 1 + 1) + (1 + (1 + 1)) = 8.
%F A000045 F(n) = Sum_{j=0..k} S(j+1,n-2j), where k = floor((n-1)/2) and the S(j,n) are the n-th j-simplex sums: S(1,n) = 1 is the 1-simplex sum, S(2,n) = Sum_{k=1..n} S(1,k) = 1+1+...+1 = n is the 2-simplex sum, S(3,n) = Sum_{k=1..n} S(2,k) = 1+2+3+...+n is the 3-simplex sum (= triangular numbers = A000217), S(4,n) = Sum_{k=1..n} S(3,k) = 1+3+6+...+n(n+1)/2 is the 4-simplex sum (= tetrahedral numbers = A000292) and so on.
%F A000045 Since S(j,n) = binomial(n-2+j,j-1), the formula above equals the well-known binomial formula, essentially. (End)
%F A000045 G.f. A(x) = x / (1 - x / (1 - x / (1 + x))). - _Michael Somos_, Jan 04 2013
%F A000045 Sum_{n >= 1} (-1)^(n-1)/(a(n)*a(n+1)) = 1/phi (phi=golden ratio). - _Vladimir Shevelev_, Feb 22 2013
%F A000045 From _Vladimir Shevelev_, Feb 24 2013: (Start)
%F A000045 (1) Expression a(n+1) via a(n): a(n+1) = (a(n) + sqrt(5*(a(n))^2 + 4*(-1)^n))/2;
%F A000045 (2) Sum_{k=1...n} (-1)^(k-1)/(a(k)*a(k+1)) = a(n)/a(n+1);
%F A000045 (3) a(n)/a(n+1) = 1/phi + r(n), where |r(n)| < 1/(a(n+1)*a(n+2)). (End)
%F A000045 F(n+1) = F(n)/2 + sqrt((-1)^n + 5*F(n)^2/4), n >= 0. F(n+1) = U_n(i/2)/i^n, (U:= Chebyshef 2nd kind). - _Bill Gosper_, Mar 04 2013
%F A000045 G.f.: -Q(0) where Q(k) = 1 - (1+x)/(1 - x/(x - 1/Q(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Mar 06 2013
%F A000045 G.f.: x-1-1/x + 1/x/Q(0), where Q(k) = 1 - (k+1)*x/(1 - x/(x - (k+1)/Q(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Apr 23 2013
%F A000045 G.f.: x*G(0), where G(k)= 1 + x*(1+x)/(1 - x*(1+x)/(x*(1+x) + 1/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jul 08 2013
%F A000045 G.f.: x^2 - 1 + 2*x^2/(W(0)-2), where W(k) = 1 + 1/(1 - x*(k + x)/( x*(k+1 + x) + 1/W(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Aug 28 2013
%F A000045 G.f.: Q(0) -1, where Q(k) = 1 + x^2 + (k+2)*x -x*(k+1 + x)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Oct 06 2013
%F A000045 Let b(n) = b(n-1) + b(n-2), with b(0) = 0, b(1) = phi. Then, for n >= 2, F(n)= floor(b(n-1)) if n is even, F(n) = ceil(b(n-1)), if n is odd, with convergence. - _Richard R. Forberg_, Jan 19 2014
%F A000045 a(n) = Sum_{t1*g(1)+t2*g(2)+...+tn*g(n)=n} multinomial(t1+t2 +...+tn,t1,t2,...,tn), where g(k)=2*k-1. - _Mircea Merca_, Feb 27 2014
%F A000045 F(n) = round(sqrt(F(n-1)^2 + F(n)^2 + F(n+1)^2)/2), for n > 0. This rule appears to apply to any sequence of the form a(n) = a(n-1) + a(n-2), for any two values of a(0) and a(1), if n is sufficiently large. - _Richard R. Forberg_, Jul 27 2014
%F A000045 F(n) = round(2/(1/F(n) + 1/F(n+1) + 1/F(n+2)), for n > 0. This rule also appears to apply to any sequence of the form a(n) = a(n-1) + a(n-2), for any two values of a(0) and a(1), if n is sufficiently large. - _Richard R. Forberg_, Aug 03 2014
%F A000045 F(n) = round(1/(Sum_{j>=n+2} 1/F(j))). - _Richard R. Forberg_, Aug 14 2014
%F A000045 a(n) = hypergeometric([-n/2+1/2, -n/2+1], [-n+1], -4) for n >= 2. - _Peter Luschny_, Sep 19 2014
%F A000045 F(n) = (L(n+1)^2 - L(n-1)^2)/(5*L(n)), where L(n) is A000032(n), with a similar inverse relationship. - _Richard R. Forberg_, Nov 17 2014
%F A000045 Consider the graph G[1-vertex;1-loop,2-loop] in comment above. Construct the power matrix array T(n,j) = [A^*j]*[S^*(j-1)] where A=(1,1,0,...) and S=(0,1,0,...)(A063524). [* is convolution operation] Define S^*0=I with I=(1,0,...). Then T(n,j) counts n-walks containing (j) loops and a(n-1) = Sum_{j=1...n} T(n,j). - _David Neil McGrath_, Nov 21 2014
%F A000045 Define F(-n) to be F(n) for n odd and -F(n) for n even. Then for all n and k, F(n) = F(k)*F(n-k+3) - F(k-1)*F(n-k+2) - F(k-2)*F(n-k) + (-1)^k*F(n-2k+2). - _Charlie Marion_, Dec 04 2014
%F A000045 F(n+k)^2 - L(k)*F(n)*F(n+k) + (-1)^k*F(n)^2 = (-1)^n*F(k)^2, if L(k) = A000032(k). - _Alexander Samokrutov_, Jul 20 2015
%F A000045 F(2*n) = F(n+1)^2 - F(n-1)^2, similar to Koshy (D) and Forberg 2011, but different. - _Hermann Stamm-Wilbrandt_, Aug 12 2015
%F A000045 F(n+1) = ceiling( (1/phi)*Sum_{k=0..n} F(k) ). - _Tom Edgar_, Sep 10 2015
%F A000045 a(n) = (L(n-3) + L(n+3))/10 where L(n)=A000032(n). - _J. M. Bergot_, Nov 25 2015
%F A000045 From _Bob Selcoe_, Mar 27 2016 (Start):
%F A000045 F(n) = (F(2n+k+1) - F(n+1)*F(n+k+1))/F(n+k), k >= 0.
%F A000045 Thus when k=0: F(n) = sqrt(F(2n+1) - F(n+1)^2).
%F A000045 F(n) = cbrt(F(3n) - F(n+1)^3 + F(n-1)^3).
%F A000045 F(n+2k) = binomial transform of any subsequence starting with F(n). Example F(6)=8: 1*8 = F(6)=8; 1*8 + 1*13 = F(8)=21; 1*8 + 2*13 + 1*21 = F(10)=55; 1*8 + 3*13 + 3*21 + 1*34 = F(12)=144, etc. This formula applies to Fibonacci-type sequences with any two seed values for a(0) and a(1) (e.g., Lucas sequence A000032: a(0)=2, a(1)=1).
%F A000045 (End)
%F A000045 F(n) = L(k)*F(n-k) + (-1)^(k+1)*F(n-2k) for all k>=0, where L(k) = A000032(k). - _Anton Zakharov_, Aug 02 2016
%F A000045 From _Ilya Gutkovskiy_, Aug 03 2016: (Start)
%F A000045 a(n) = F_n(1), where F_n(x) are the Fibonacci polynomials.
%F A000045 Inverse binomial transform of A001906.
%F A000045 Number of zeros in substitution system {0 -> 11, 1 -> 1010} at step n from initial string "1" (1 -> 1010 -> 101011101011 -> ...) multiplied by 1/A000079(n). (End)
%F A000045 For n>=2, a(n) = 2^(n^2+n) - (4^n-2^n-1)*floor(2^(n^2+n)/(4^n-2^n-1)) - 2^n*floor(2^(n^2) - (2^n-1-1/2^n)*floor(2^(n^2+n)/(4^n-2^n-1))). - _Benoit Cloitre_, Apr 17 2017
%F A000045 For n > 0, a(n) = b(n+1) where b(n) = Sum_{k=1..n} b(n-k)*A000931(k-1), b(0) = 1. - _J. Conrad_, Apr 19 2017
%F A000045 f(n+1) = Sum_{j=0..floor(n/2)} Sum_{k=0..j} binomial(n-2j,k)*binomial(j,k). - _Tony Foster III_, Sep 04 2017
%F A000045 F(n) = Sum_{k=0..floor((n-1)/2)} ( (n-k-1)! / ((n-2k-1)! * k!) ). - _Zhandos Mambetaliyev_, Nov 08 2017
%F A000045 For x even, F(n) = (F(n+x) + F(n-x))/L(x). For x odd, F(n) = (F(n+x) - F(n-x))/L(x) where n>=x in both cases. Therefore F(n) = F(2*n)/L(n) for n>=0. - _David James Sycamore_, May 04 2018
%F A000045 From _Isaac Saffold_, Jul 19 2018: (Start)
%F A000045 Let [a/p] denote the Legendre symbol. Then, for an odd prime p:
%F A000045   F(p+n) == [5/p]*F([5/p]+n) (mod p), if [5/p] = 1 or -1.
%F A000045   F(p+n) == 3*F(n) (mod p), if [5/p] = 0 (i.e. p = 5).
%F A000045   This is true for negative-indexed terms as well, if this sequence is extended by the negafibonacci numbers (i.e. F(-n) = A039834(n)). (End)
%F A000045 a(n) = A094718(4, n). a(n) = A101220(0, j, n).
%F A000045 a(n) = A090888(0, n+1) = A118654(0, n+1) = A118654(1, n-1) = A109754(0, n) = A109754(1, n-1), for n > 0.
%F A000045 a(n) = (L(n-3) + L(n-2) + L(n-1) + L(n))/5 with L(n)=A000032(n). - _Art Baker_, Jan 04 2019
%e A000045 For x = 0,1,2,3,4, x=1/(x+1) = 1, 1/2, 2/3, 3/5, 5/8. These fractions have numerators 1,1,2,3,5, which are the 2nd to 6th entries in the sequence. - _Cino Hilliard_, Sep 15 2008
%e A000045 From _Joerg Arndt_, May 21 2013: (Start)
%e A000045 There are a(7)=13 compositions of 7 where there is a drop between every second pair of parts, starting with the first and second part:
%e A000045 01:  [ 2 1 2 1 1 ]
%e A000045 02:  [ 2 1 3 1 ]
%e A000045 03:  [ 2 1 4 ]
%e A000045 04:  [ 3 1 2 1 ]
%e A000045 05:  [ 3 1 3 ]
%e A000045 06:  [ 3 2 2 ]
%e A000045 07:  [ 4 1 2 ]
%e A000045 08:  [ 4 2 1 ]
%e A000045 09:  [ 4 3 ]
%e A000045 10:  [ 5 1 1 ]
%e A000045 11:  [ 5 2 ]
%e A000045 12:  [ 6 1 ]
%e A000045 13:  [ 7 ]
%e A000045 There are abs(a(6+1))=13 compositions of 6 where there is no rise between every second pair of parts, starting with the second and third part:
%e A000045 01:  [ 1 2 1 2 ]
%e A000045 02:  [ 1 3 1 1 ]
%e A000045 03:  [ 1 3 2 ]
%e A000045 04:  [ 1 4 1 ]
%e A000045 05:  [ 1 5 ]
%e A000045 06:  [ 2 2 1 1 ]
%e A000045 07:  [ 2 3 1 ]
%e A000045 08:  [ 2 4 ]
%e A000045 09:  [ 3 2 1 ]
%e A000045 10:  [ 3 3 ]
%e A000045 11:  [ 4 2 ]
%e A000045 12:  [ 5 1 ]
%e A000045 13:  [ 6 ]
%e A000045 (End)
%e A000045 Partially ordered partitions of (n-1) into parts 1,2,3 where only the order of the adjacent 1's and 2's are unimportant. E.g., a(8)=21. These are (331),(313),(133),(322),(232),(223),(3211),(2311),(1321),(2131),(1132),(2113),(31111),(13111),(11311),(11131),(11113),(2221),(22111),(211111),(1111111). - _David Neil McGrath_, Jul 25 2015
%e A000045 Consider the partitions of 7 with summands initially listed in nonincreasing order. Keep the 1's frozen in position,(indicated by "[]") and then allow the other summands to otherwise vary their order: 7; 6,[1]; 5,2; 2,5; 4,3; 3,4; 5,[1,1], 4,2,[1]; 2,4,[1]; 3,3,[1]; 3,3,2; 3,2,3; 2,3,3; 4,[1,1,1]; 3,2,[1,1]; 2,3,[1,1]; 2,2,2,[1]; 3,[1,1,1,1]; 2,2,[1,1,1]; 2,[1,1,1,1,1]; [1,1,1,1,1,1,1]. There are 21 = a(7+1) arrangements in all. - _Gregory L. Simay_, Jun 14 2016
%p A000045 A000045 := proc(n) combinat[fibonacci](n); end;
%p A000045 ZL:=[S, {a = Atom, b = Atom, S = Prod(X,Sequence(Prod(X,b))), X = Sequence(b,card >= 1)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=0..38); # _Zerinvary Lajos_, Apr 04 2008
%p A000045 spec := [B, {B=Sequence(Set(Z, card>1))}, unlabeled ]: seq(combstruct[count](spec, size=n), n=1..39); # _Zerinvary Lajos_, Apr 04 2008
%p A000045 # The following Maple command isFib(n) yields true or false depending on whether n is a Fibonacci number or not.
%p A000045 with(combinat): isFib := proc(n) local a: a := proc(n) local j: for j while fibonacci(j) <= n do fibonacci(j) end do: fibonacci(j-1) end proc: evalb(a(n) = n) end proc: # _Emeric Deutsch_, Nov 11 2014
%t A000045 Table[Fibonacci[k], {k, 0, 50}] (* _Mohammad K. Azarian_, Jul 11 2015 *)
%t A000045 Table[2^n Sqrt @ Product[(Cos[Pi k/(n + 1)]^2 + 1/4), {k, n}] // FullSimplify, {n, 15}]; (* Kasteleyn's formula specialized, Sarah-Marie Belcastro (smbelcas(AT)toroidalsnark.net), Jul 04 2009 *)
%t A000045 LinearRecurrence[{1, 1}, {0, 1}, 40] (* _Harvey P. Dale_, Aug 03 2014 *)
%t A000045 Fibonacci[Range[0, 20]] (* _Eric W. Weisstein_, Sep 22 2017 *)
%t A000045 CoefficientList[Series[-(x/(-1 + x + x^2)), {x, 0, 20}], x] (* _Eric W. Weisstein_, Sep 22 2017 *)
%o A000045 (Axiom) [fibonacci(n) for n in 0..50]
%o A000045 (MAGMA) [Fibonacci(n): n in [0..38]];
%o A000045 (MAGMA) [0,1] cat [n: n in [1..50000000] | IsSquare(5*n^2-4) or IsSquare(5*n^2+4)]; // _Vincenzo Librandi_, Nov 19 2014
%o A000045 (Maxima) makelist(fib(n),n,0,100); /* _Martin Ettl_, Oct 21 2012 */
%o A000045 (PARI) a(n) = fibonacci(n)
%o A000045 (PARI) a(n) = imag(quadgen(5)^n)
%o A000045 (PARI) a(n)=my(phi=quadgen(5));(phi^n-(-1/phi)^n)/(2*phi-1) \\ _Charles R Greathouse IV_, Jun 17 2012
%o A000045 (PARI) a(n)=polcoeff(sum(m=0, n, x^m*prod(k=1, m, k+x +x*O(x^n))/prod(k=1, m, 1+k*x +x*O(x^n))), n) \\ _Paul D. Hanna_, Oct 26 2013
%o A000045 (Python) # _Jaap Spies_, Jan 05 2007 (Change leading dots to blanks.)
%o A000045 def fib():
%o A000045 ... """ Generates the Fibonacci numbers, starting with 0 """
%o A000045 ... x, y = 0, 1
%o A000045 ... while 1:
%o A000045 ....... yield x
%o A000045 ....... x, y = y, x+y
%o A000045 .
%o A000045 f = fib()
%o A000045 a = [f.next() for i in range(100)]
%o A000045 .
%o A000045 def A000045(n):
%o A000045 ... """ Returns Fibonacci number with index n, offset 0,4 """
%o A000045 ... return a[n]
%o A000045 ................
%o A000045 def A000045_list(N):
%o A000045 ... """ Returns a list of the first n Fibonacci numbers """
%o A000045 ... return a[:N]
%o A000045 .
%o A000045 (Python) # As b-file:
%o A000045 from gmpy2 import fib
%o A000045 for n in xrange(100): print str(n)+" "+str(fib(n)) # _Bruno Berselli_, Dec 06 2016
%o A000045 (Sage) ## Demonstration program from Jaap Spies:
%o A000045 a = sloane.A000045; ## choose sequence
%o A000045 print a ## This returns the name of the sequence.
%o A000045 print a(38) ## This returns the 38th number of the sequence.
%o A000045 print a.list(39) ## This returns a list of the first 39 numbers.
%o A000045 (Sage) # Alternatively:
%o A000045 a = BinaryRecurrenceSequence(1,1); print [a(n) for n in (0..19)]
%o A000045 # Closed form integer formula with F(1) = 0 from Paul Hankin (use only for fun).
%o A000045 F = lambda n: (4<<(n-1)*(n+2))//((4<<2*(n-1))-(2<<(n-1))-1)&((2<<(n-1))-1)
%o A000045 print [F(n) for n in (0..19)] # _Peter Luschny_, Aug 28 2016
%o A000045 (Sage) [i for i in fibonacci_sequence(0, 40)] # _Bruno Berselli_, Jun 26 2014
%o A000045 (Haskell)
%o A000045 -- Based on code from http://www.haskell.org/haskellwiki/The_Fibonacci_sequence
%o A000045 -- which also has other versions.
%o A000045 fib :: Int -> Integer
%o A000045 fib n = fibs !! n
%o A000045 .. where
%o A000045 .... fibs = 0 : 1 : zipWith (+) fibs (tail fibs)
%o A000045 {- Example of use: map fib [0..38] _Gerald McGarvey_, Sep 29 2009 -}
%o A000045 (Julia)
%o A000045 function fib(n)
%o A000045    F = BigInt[1 1; 1 0]
%o A000045    Fn = F^n
%o A000045    Fn[2, 1]
%o A000045 end
%o A000045 println([fib(n) for n in 0:38]) # _Peter Luschny_, Feb 23 2017
%o A000045 (GAP)
%o A000045 Fib:=[0,1];; for n in [3..10^3] do Fib[n]:=Fib[n-1]+Fib[n-2]; od; Fib;  #  _Muniru A Asiru_, Sep 03 2017
%o A000045 (Scheme)
%o A000045 ;; The following definition uses macro definec for the memoization (caching) of the results. See http://oeis.org/wiki/Memoization#Scheme
%o A000045 (definec (A000045 n) (if (< n 2) n (+ (A000045 (- n 1)) (A000045 (- n 2))))) ;; _Antti Karttunen_, Oct 06 2017
%o A000045 (Scala) def fibonacci(n: BigInt): BigInt = {
%o A000045   val zero = BigInt(0)
%o A000045   def fibTail(n: BigInt, a: BigInt, b: BigInt): BigInt = n match {
%o A000045     case `zero` => a
%o A000045     case _ => fibTail(n - 1, b, a + b)
%o A000045   }
%o A000045   fibTail(n, 0, 1)
%o A000045 } // Based on "Case 3: Tail Recursion" from Carrasquel (2016) link
%o A000045 (0 to 49).map(fibonacci(_)) // _Alonso del Arte_, Apr 13 2019
%Y A000045 Cf. A039834 (signed Fibonacci numbers), A001690 (complement), A000213, A000288, A000322, A000383, A060455, A030186, A020695, A020701, A071679, A099731, A100492, A094216, A094638, A000108, A101399, A101400, A001611, A000071, A157725, A001911, A157726, A006327, A157727, A157728, A157729, A167616, A059929, A144152, A152063, A114690, A003893, A000032, A060441, A000930, A003269, A000957, A057078, A007317, A091867, A104597, A249548, A262342, A001060, A022095, A072649.
%Y A000045 First row of arrays A103323, A172236, A234357. Second row of arrays A099390, A048887, and A092921 (k-generalized Fibonacci numbers).
%Y A000045 Cf. A001175 (Pisano periods), A001177 (Entry points), A001176 (number of zeros in a fundamental period).
%Y A000045 Fibonacci-Pascal triangles: A027926, A036355, A037027, A074829, A105809, A109906, A111006, A114197, A162741, A228074.
%Y A000045 Boustrophedon transforms: A000738, A000744.
%Y A000045 Powers: A103323, A105317, A254719.
%Y A000045 Numbers of prime factors: A022307 and A038575.
%Y A000045 Cf. A163733.
%K A000045 nonn,core,nice,easy,hear
%O A000045 0,4
%A A000045 _N. J. A. Sloane_, 1964

%I A290689
%S A290689 1,1,1,2,3,5,8,13,21,34,55,88,143,229,370,592,955,1527,2457,3929
%N A290689 Number of transitive rooted trees with n nodes.
%C A290689 A rooted tree is transitive if every proper terminal subtree is also a branch of the root. First differs from A206139 at a(13) = 143.
%e A290689 The a(7) = 8 trees are: (o(oooo)), (oo(ooo)), (o(o)((o))), (o(o)(oo)), (ooo(oo)), (oo(o)(o)), (oooo(o)), (oooooo).
%t A290689 nn=18;
%t A290689 rtall[n_]:=If[n===1,{{}},Module[{cas},Union[Sort/@Join@@(Tuples[rtall/@#]&/@IntegerPartitions[n-1])]]];
%t A290689 Table[Length[Select[rtall[n],Complement[Union@@#,#]==={}&]],{n,nn}]
%Y A290689 Cf. A000081, A004111, A279861, A290822.
%K A290689 nonn,more
%O A290689 1,4
%A A290689 _Gus Wiseman_, Oct 19 2017
%E A290689 a(20) from _Robert Price_, Sep 13 2018

%I A027926
%S A027926 1,1,1,1,1,1,2,2,1,1,1,2,3,4,3,1,1,1,2,3,5,7,7,4,1,1,1,2,3,5,8,12,14,
%T A027926 11,5,1,1,1,2,3,5,8,13,20,26,25,16,6,1,1,1,2,3,5,8,13,21,33,46,51,41,
%U A027926 22,7,1,1,1,2,3,5,8,13,21,34,54,79,97,92,63,29
%N A027926 Triangular array T read by rows: T(n,0) = T(n,2n) = 1 for n >= 0; T(n,1) = 1 for n >= 1; T(n,k) = T(n-1,k-2) + T(n-1,k-1) for k = 2..2n-1, n >= 2.
%C A027926 T(n,k) = number of strings s(0),...,s(n) such that s(0)=0, s(n)=n-k and for 1<=i<=n, s(i)=s(i-1)+d, with d in {0,1,2} if i=0, in {0,2} if s(i)=2i, in {0,1,2} if s(i)=2i-1, in {0,1} if 0<=s(i)<=2i-2.
%C A027926 Can be seen as concatenation of triangles A104763 and A105809, with identifying column of Fibonacci numbers, see example. - _Reinhard Zumkeller_, Aug 15 2013
%H A027926 Reinhard Zumkeller, <a href="/A027926/b027926.txt">Rows n = 0..100 of table, flattened</a>
%H A027926 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A027926 T(n, k)=sum(binomial(n-j, 2n-k-2j), j=0..floor[(2n-k+1)/2]). - _Len Smiley_, Oct 21 2001
%e A027926 .   0:                           1
%e A027926 .   1:                        1  1   1
%e A027926 .   2:                     1  1  2   2   1
%e A027926 .   3:                  1  1  2  3   4   3   1
%e A027926 .   4:               1  1  2  3  5   7   7   4   1
%e A027926 .   5:            1  1  2  3  5  8  12  14  11   5   1
%e A027926 .   6:          1 1  2  3  5  8 13  20  26  25  16   6   1
%e A027926 .   7:        1 1 2  3  5  8 13 21  33  46  51  41  22   7   1
%e A027926 .   8:      1 1 2 3  5  8 13 21 34  54  79  97  92  63  29   8  1
%e A027926 .   9:    1 1 2 3 5  8 13 21 34 55  88 133 176 189 155  92  37  9  1
%e A027926 .  10:  1 1 2 3 5 8 13 21 34 55 89 143 221 309 365 344 247 129 46 10  1
%e A027926 .
%e A027926 .   1:                           1
%e A027926 .   2:                        1  1
%e A027926 .   3:                     1  1  2
%e A027926 .   4:                  1  1  2  3
%e A027926 .   5:               1  1  2  3  5      columns = A000045, > 0
%e A027926 .   6:            1  1  2  3  5  8     +---------+
%e A027926 .   7:          1 1  2  3  5  8 13     | A104763 |
%e A027926 .   8:        1 1 2  3  5  8 13 21     +---------+
%e A027926 .   9:      1 1 2 3  5  8 13 21 34
%e A027926 .  10:    1 1 2 3 5  8 13 21 34 55
%e A027926 .  11:  1 1 2 3 5 8 13 21 34 55 89
%e A027926 .
%e A027926 .   0:                           1
%e A027926 .   1:                           1   1                +---------+
%e A027926 .   2:                           2   2   1            | A105809 |
%e A027926 .   3:                           3   4   3   1        +---------+
%e A027926 .   4:                           5   7   7   4   1
%e A027926 .   5:                           8  12  14  11   5   1
%e A027926 .   6:                          13  20  26  25  16   6   1
%e A027926 .   7:                          21  33  46  51  41  22   7   1
%e A027926 .   8:                          34  54  79  97  92  63  29   8  1
%e A027926 .   9:                          55  88 133 176 189 155  92  37  9  1
%e A027926 .  10:                          89 143 221 309 365 344 247 129 46 10  1
%p A027926 A027926 := proc(n,k)
%p A027926     add(binomial(n-j,2*n-k-2*j),j=0..(2*n-k+1)/2) ;
%p A027926 end proc: # _R. J. Mathar_, Apr 11 2016
%t A027926 z = 15; t[n_, 0] := 1; t[n_, k_] := 1 /; k == 2 n; t[n_, 1] := 1;
%t A027926 t[n_, k_] := t[n, k] = t[n - 1, k - 2] + t[n - 1, k - 1];
%t A027926 u = Table[t[n, k], {n, 0, z}, {k, 0, 2 n}];
%t A027926 TableForm[u] (* A027926 array *)
%t A027926 v = Flatten[u] (* A027926 sequence *)
%t A027926 (* _Clark Kimberling_, Aug 31 2014 *)
%o A027926 (PARI) {T(n, k) = if( k<0 || k>2*n, 0, if( k<=1 || k==2*n, 1, T(n-1, k-2) + T(n-1, k-1)))}; /* __Michael Somos_, Feb 26 1999 */
%o A027926 (PARI) {T(n, k) = if( k<0 || k>2*n, 0, sum( j=max(0, k-n), k\2, binomial(k-j, j)))}; /* _Michael Somos_ */
%o A027926 (Haskell)
%o A027926 a027926 n k = a027926_tabf !! n !! k
%o A027926 a027926_row n = a027926_tabf !! n
%o A027926 a027926_tabf = iterate (\xs -> zipWith (+)
%o A027926                                ([0] ++ xs ++ [0]) ([1,0] ++ xs)) [1]
%o A027926 -- Variant, cf. example:
%o A027926 a027926_tabf' = zipWith (++) a104763_tabl (map tail a105809_tabl)
%o A027926 -- _Reinhard Zumkeller_, Aug 15 2013
%Y A027926 Many columns of T are A000045 (Fibonacci sequence), also in T: A001924, A004006, A000071, A000124, A014162, A014166, A027927-A027933.
%Y A027926 Some other Fibonacci-Pascal triangles: A036355, A037027, A074829, A105809, A109906, A111006, A114197, A162741, A228074.
%K A027926 nonn,tabf
%O A027926 0,7
%A A027926 _Clark Kimberling_
%E A027926 Incorporates comments from _Michael Somos_.
%E A027926 Example extended by _Reinhard Zumkeller_, Aug 15 2013

%I A014260
%S A014260 0,1,1,2,3,5,8,13,21,52,64,89,135,233,764,1096,1563,8464,12115,16763,
%T A014260 67884,104645,153521,699922,825273,1055269,1427797,11053298,19030539,
%U A014260 108265550,201768641,257331442,404198544,648332296,1094223700
%N A014260 Iccanobif numbers: add a(n-1) to reversal of a(n-2).
%H A014260 Alois P. Heinz, <a href="/A014260/b014260.txt">Table of n, a(n) for n = 0..1000</a>
%p A014260 R:= n-> (s-> parse(cat(s[-i]$i=1..length(s))))(""||n):
%p A014260 a:= proc(n) option remember; `if`(n<2, n,
%p A014260        a(n-1) +R(a(n-2)))
%p A014260     end:
%p A014260 seq(a(n), n=0..50);  # _Alois P. Heinz_, Jun 18 2014
%t A014260 Clear[ Bif ]; Bif[ 0 ]=0; Bif[ 1 ]=1; Bif[ n_Integer ] := Bif[ n ]=Bif[ n-1 ]+Plus@@(IntegerDigits[ Bif[ n-2 ], 10 ]//(#*Array[ 10^#&, Length[ # ], 0 ])&); Array[ Bif, 40, 0 ]
%t A014260 nxt[{a_,b_}]:={b,IntegerReverse[a]+b}; NestList[nxt,{0,1},40][[All,1]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Jul 04 2018 *)
%Y A014260 Cf. A000045, A001129, A014258, A014259.
%Y A014260 See A000045 for the Fibonacci numbers.
%K A014260 base,nonn,easy
%O A014260 0,4
%A A014260 _N. J. A. Sloane_.

%I A181600
%S A181600 1,1,2,3,5,8,13,21,33,53,85,136,218,349,559,895,1434,2297,3679,5893,
%T A181600 9439,15119,24217,38790,62132,99520,159407,255331,408978,655083,
%U A181600 1049283,1680695,2692063,4312028,6906816,11063033,17720278,28383559,45463532,72821479
%N A181600 Expansion of 1/(1 - x - x^2 + x^8 - x^10).
%C A181600 Limiting ratio is 1.60176..., the largest real root of -1 + x^2 - x^8 - x^9 + x^10. Compare this constant to Lehmer's Salem constant A073011 and the golden mean.
%H A181600 G. C. Greubel, <a href="/A181600/b181600.txt">Table of n, a(n) for n = 0..1000</a>
%H A181600 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,0,0,0,0,-1,0,1).
%F A181600 a(n) = a(n-1) + a(n-2) - a(n-8) + a(n-10). - _Franck Maminirina Ramaharo_, Oct 31 2018
%t A181600 CoefficientList[Series[1/(1 - x - x^2 + x^8 - x^10), {x, 0, 50}], x]
%t A181600 LinearRecurrence[{1, 1, 0, 0, 0, 0, 0, -1, 0, 1}, {1, 1, 2, 3, 5, 8, 13, 21, 33, 53}, 50] (* _Harvey P. Dale_, Aug 11 2015 *)
%o A181600 (PARI) Vec(1/(1 -x -x^2 +x^8 -x^10) + O(x^50)) \\ _G. C. Greubel_, Nov 16 2016
%o A181600 (MAGMA) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1 -x-x^2+x^8-x^10))); // _G. C. Greubel_, Nov 03 2018
%Y A181600 Cf. A029826, A117791, A143419, A143438, A143472, A143619, A143644, A147663, A173908, A173911, A173924, A173925, A174522, A175740, A175772, A175773, A175782, A204631, A225391, A225393, A225394, A225482, A225499.
%K A181600 nonn,easy
%O A181600 0,3
%A A181600 _Roger L. Bagula_, May 06 2013

%I A212804
%S A212804 1,0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,
%T A212804 6765,10946,17711,28657,46368,75025,121393,196418,317811,514229,
%U A212804 832040,1346269,2178309,3524578,5702887,9227465,14930352,24157817,39088169,63245986,102334155,165580141,267914296,433494437,701408733,1134903170,1836311903,2971215073,4807526976
%N A212804 Expansion of (1-x)/(1-x-x^2).
%C A212804 A variant of the Fibonacci number A000045.
%C A212804 Number of compositions of n into parts >= 2. - _Joerg Arndt_, Aug 13 2012
%C A212804 From _Petros Hadjicostas_, Jan 08 2019: (Start)
%C A212804 For n >= 0, a(n) is the number of unmarked circular binary words (necklaces) of length n+1 with exactly one occurrence of the pattern 00 (provided we allow the strings of length 1, i.e., 0 and 1, to wrap around themselves on a circle to form strings of length 2). See the comments for array A320341.
%C A212804 Using MacMahon's bijection between necklaces and cyclic compositions, we conclude that a(n) is also the number of (unmarked) cyclic compositions of n+1 with exactly one 1.
%C A212804 Removing the single 1 from each cyclic composition of n+1, we get all linear compositions of n with each part >= 2, which is what is stated above by Joerg Arndt.
%C A212804 (End)
%H A212804 Vincenzo Librandi, <a href="/A212804/b212804.txt">Table of n, a(n) for n = 0..1000</a>
%H A212804 J. J. Madden, <a href="http://arxiv.org/abs/1707.04351">A generating function for the distribution of runs in binary words</a>, arXiv:1707.04351 [math.CO], 2017, Theorem 1.1, r=1 and k=0.
%H A212804 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1).
%F A212804 G.f.: (1-x)/(1-x-x^2) = (1-x)*G(0)/(x*sqrt(5)) where G(k)= 1 -((-1)^k)*2^k/(a^k - b*x*a^k*2^k/(b*x*2^k - 2*((-1)^k)*c^k/G(k+1))) and a=3+sqrt(5), b=1+sqrt(5), c=3-sqrt(5); (continued fraction, 3rd kind, 3-step). - _Sergei N. Gladkovskii_, Jun 04 2012
%F A212804 G.f.: 1/(1-(Sum_{k>=2} x^k)). - _Joerg Arndt_, Aug 13 2012
%F A212804 a(n) = Fibonacci(n+1) - Fibonacci(n). - _Arkadiusz Wesolowski_, Oct 29 2012
%F A212804 G.f.: 1 - x*Q(0) where Q(k) = 1 - (1+x)/(1 - x/(x - 1/Q(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Mar 06 2013
%F A212804 G.f.: 3*x^3/(3*x - Q(0)) - x^2 + 1, where Q(k) = 1 - 1/(4^k - x*16^k/(x*4^k - 1/(1 + 1/(2*4^k - 4*x*16^k/(2*x*4^k +1/Q(k+1)))))); (continued fraction). - _Sergei N. Gladkovskii_, May 21 2013
%F A212804 G.f.: G(0)*(1-x)/(2-x), where G(k)= 1 + 1/(1 - (x*(5*k-1))/((x*(5*k+4)) - 2/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 15 2013
%F A212804 G.f.: 1 + Q(0)*x^2/2, where Q(k) = 1 + 1/(1 - x*(2*k+1 + x)/( x*(2*k+2 + x) + 1/Q(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Aug 29 2013
%F A212804 a(n) = Sum_{k=0..n} (C(k,n-k) - C(k,n-k-1)). - _Peter Luschny_, Oct 01 2014
%F A212804 a(n) = (2^(-1-n)*((1-sqrt(5))^n*(1+sqrt(5))+(-1+sqrt(5))*(1+sqrt(5))^n))/sqrt(5). - _Colin Barker_, Sep 25 2016
%F A212804 a(n) = A000045(n-1), n>=1. - _R. J. Mathar_, Apr 14 2018
%e A212804 From _Petros Hadjicostas_, Jan 08 2019: (Start)
%e A212804 For n=6, we have a(6) = 5. The binary necklaces of length n+1 = 7 with exactly one occurrence of 00 are as follows: 0011111, 0010111, 0011011, 0011101, and 0010101.
%e A212804 The corresponding cyclic compositions of n+1 = 7 with exactly one 1 (under MacMahon's bijection) are as follows: 1+6, 1+2+4, 1+3+3, 1+4+2, 1+2+2+2.
%e A212804 Of course, removing the 1 from the cyclic composition, we get a (linear) composition of n=6 with parts >=2 (as stated above by Joerg Arndt): 6, 2+4, 3+3, 4+2, 2+2+2. (For linear compositions, 2+4 is not the same as 4+2.)
%e A212804 (End)
%t A212804 Table[Fibonacci[n-1], {n, 0, 40}] (* _Vladimir Reshetnikov_, Sep 24 2016 *)
%o A212804 (MAGMA) [Fibonacci(n + 1) - Fibonacci(n): n in [0..50]]; // _Vincenzo Librandi_, Dec 09 2012
%Y A212804 Cf. A000045, A105809 (alternating row sums).
%Y A212804 Column k=1 of A320341.
%K A212804 nonn,easy
%O A212804 0,5
%A A212804 _N. J. A. Sloane_, May 27 2012, following a suggestion from _R. K. Guy_.

%I A104763
%S A104763 1,1,1,1,1,2,1,1,2,3,1,1,2,3,5,1,1,2,3,5,8,1,1,2,3,5,8,13,1,1,2,3,5,8,
%T A104763 13,21,1,1,2,3,5,8,13,21,34,1,1,2,3,5,8,13,21,34,55,1,1,2,3,5,8,13,21,
%U A104763 34,55,89,1,1,2,3,5,8,13,21,34,55,89,144,1,1,2,3,5,8,13,21,34,55,89,144
%N A104763 Triangle read by rows: Fibonacci(1), Fibonacci(2), ..., Fibonacci(n) in row n.
%C A104763 Triangle of A104762, Fibonacci sequence in each row starts from the right.
%C A104763 The triangle or chess sums, see A180662 for their definitions, link the Fibonacci(n) triangle to sixteen different sequences, see the crossrefs. The knight sums Kn14 - Kn18 have been added. As could be expected all sums are related to the Fibonacci numbers. - _Johannes W. Meijer_, Sep 22 2010
%C A104763 Sequence B is called a reluctant sequence of sequence A, if B is triangle array read by rows: row number k coincides with first k elements of the sequence A. Sequence A104763 is reluctant sequence of Fibonacci numbers (A000045), except 0. - _Boris Putievskiy_, Dec 13 2012
%H A104763 Reinhard Zumkeller, <a href="/A104763/b104763.txt">Rows n = 1..100 of table, flattened</a>
%H A104763 Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations Integer Sequences And Pairing Functions</a> arXiv:1212.2732 [math.CO]
%F A104763 F(1) through F(n) starting from the left in n-th row.
%F A104763 T(n,k)=A000045(k), 1<=k<=n. - _R. J. Mathar_, May 02 2008
%F A104763 a(n) = A000045(m), where m= n-t(t+1)/2, t=floor((-1+sqrt(8*n-7))/2). - _Boris Putievskiy_, Dec 13 2012
%e A104763 First few rows of the triangle are:
%e A104763 1;
%e A104763 1, 1;
%e A104763 1, 1, 2;
%e A104763 1, 1, 2, 3;
%e A104763 1, 1, 2, 3, 5;
%e A104763 1, 1, 2, 3, 5, 8;
%e A104763 1, 1, 2, 3, 5, 8, 13;
%e A104763 ...
%o A104763 (Haskell)
%o A104763 a104763 n k = a104763_tabl !! (n-1) !! (k-1)
%o A104763 a104763_row n = a104763_tabl !! (n-1)
%o A104763 a104763_tabl = map (flip take $ tail a000045_list) [1..]
%o A104763 -- _Reinhard Zumkeller_, Aug 15 2013
%Y A104763 Cf. A104762, A000045.
%Y A104763 Cf. A000071 (row sums). - _R. J. Mathar_, Jul 22 2009
%Y A104763 Triangle sums (see the comments): A000071 (Row1; Kn4 & Ca1 & Ca4 & Gi1 & Gi4); A008346 (Row2); A131524 (Kn11); A001911 (Kn12); A006327 (Kn13); A167616 (Kn14); A180671 (Kn15); A180672 (Kn16); A180673 (Kn17); A180674 (Kn18); A052952 (Kn21 & Kn22 & Kn23 & Fi2 & Ze2); A001906 (Kn3 &Fi1 & Ze3); A004695 (Ca2 & Ze4); A001076 (Ca3 & Ze1); A080239 (Gi2); A081016 (Gi3). - _Johannes W. Meijer_, Sep 22 2010
%K A104763 nonn,tabl,easy
%O A104763 1,6
%A A104763 _Gary W. Adamson_, Mar 23 2005
%E A104763 Edited by _R. J. Mathar_, May 02 2008
%E A104763 Extended by _R. J. Mathar_, Aug 27 2008

%I A240733
%S A240733 1,1,2,3,5,8,13,21,32,50,78,121,187,289,448,693,1072,1658,2564,3966,
%T A240733 6134,9487,14673,22695,35101,54288,83964,129862,200850,310643,480452,
%U A240733 743085,1149282,1777523,2749182,4251987,6576279,10171116,15731022,24330178,37629950
%N A240733 Floor(6^n/(2+2*cos(Pi/9))^n).
%C A240733 a(n) is the perimeter (rounded down) of a nonaflake after n iterations, let a(0) = 1. The total number of sides is 9*A000400(n). The total number of holes is A002452(n). 2*cos(Pi/9) = 1.87938524... = diagonal b of nonagon (see comments in A123609).
%H A240733 Vincenzo Librandi, <a href="/A240733/b240733.txt">Table of n, a(n) for n = 0..1000</a>
%H A240733 Kival Ngaokrajang, <a href="/A240733/a240733_1.pdf">Illustration of nonaflake for n = 0..3</a>
%H A240733 Wikipedia, <a href="http://en.wikipedia.org/wiki/N-flake">n-flake</a>
%p A240733 A240733:=n->floor(6^n/(2+2*cos(Pi/9))^n); seq(A240733(n), n=0..50); # _Wesley Ivan Hurt_, Apr 12 2014
%t A240733 Table[Floor[6^n/(2 + 2*Cos[Pi/9])^n], {n, 0, 50}] (* _Wesley Ivan Hurt_, Apr 12 2014 *)
%o A240733 (PARI) {a(n)=floor(6^n/(2+2*cos(Pi/9))^n)}
%o A240733        for (n=0, 100, print1(a(n), ", "))
%Y A240733 Cf. A000400, A002452, A123609, A240523 (pentaflake), A240671 (heptaflake), A240572 (octaflake), A240733 (nonaflake), A240734 (decaflake), A240735 (dodecaflake).
%K A240733 nonn,easy
%O A240733 0,3
%A A240733 _Kival Ngaokrajang_, Apr 11 2014

%I A177194
%S A177194 1,1,2,3,5,8,13,21,34,55,89,144,233,377,987,1597,2584,4181,6765,17711,
%T A177194 28657,46368,121393,196418,317811,514229,1346269,3524578,9227465,
%U A177194 24157817,63245986,267914296,433494437,53316291173,86267571272
%N A177194 Fibonacci numbers whose decimal expansion does not contain any digit 0.
%C A177194 The probability that fib(n) contains no 0's decreases to zero as n goes to infinity. Its maximum value seems to be F(184) having 39 digits, including no zeros.
%F A177194 a(n) = A000045(A076564(n)). [From _R. J. Mathar_, Oct 18 2010]
%e A177194 a(7)=13 since fib(7) does not contain the digit 0.
%Y A177194 Cf. A176253, A000045
%K A177194 nonn,base
%O A177194 1,3
%A A177194 _Carmine Suriano_, May 04 2010

%I A005181 M0693
%S A005181 1,1,2,3,5,8,13,21,34,55,91,149,245,404,666,1097,1809,2981,4915,8104,
%T A005181 13360,22027,36316,59875,98716,162755,268338,442414,729417,1202605,
%U A005181 1982760,3269018,5389699,8886111,14650720,24154953,39824785,65659970,108254988,178482301
%N A005181 a(n) = ceiling(exp((n-1)/2)).
%C A005181 This sequence illustrates the second law of small numbers because it is a coincidence that its first ten terms are the same as the first ten Fibonacci numbers (A000045). - _Alonso del Arte_, Mar 18 2013
%D A005181 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A005181 I. Stewart, L'univers des nombres, pp. 27 Belin-Pour La Science, Paris 2000.
%H A005181 Vladimir Pletser, <a href="/A005181/b005181.txt">Table of n, a(n) for n = 0..1000</a>
%H A005181 R. K. Guy, <a href="/A005347/a005347.pdf">The Second Strong Law of Small Numbers</a>, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
%H A005181 R. K. Guy, <a href="http://www.jstor.org/stable/2691503">The Second Strong Law of Small Numbers</a>, Math. Mag, 63 (1990), no. 1, 3-20.
%H A005181 R. K. Guy and N. J. A. Sloane, <a href="/A005180/a005180.pdf">Correspondence</a>, 1988.
%H A005181 I. Stewart, <a href="http://www.whydomath.org/Reading_Room_Material/ian_stewart/9505.html">Fibonacci Forgeries</a>
%H A005181 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number.</a>
%H A005181 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StrongLawofSmallNumbers.html">Strong Law of Small Numbers.</a>
%F A005181 Lim_{n->oo} a(n+1)/a(n) = sqrt(e) = 1.64872127... = A019774. - _Alois P. Heinz_, Feb 19 2019
%p A005181 seq(round(ceil(exp((n-1)/2))), n=0..50); # _Vladimir Pletser_, Sep 15 2013;
%t A005181 Table[Ceiling[E^((n - 1)/2)], {n, 0, 39}] (* _Alonso del Arte_, Mar 18 2013 *)
%o A005181 (Python)
%o A005181 import math
%o A005181 for n in range(99):
%o A005181   print str(int(math.ceil(math.e**((n-1)*0.5))))+',',
%o A005181 # _Alex Ratushnyak_, Mar 18 2013
%Y A005181 Cf. A000045, A019774.
%K A005181 nonn
%O A005181 0,3
%A A005181 _N. J. A. Sloane_, _R. K. Guy_
%E A005181 A few more terms from _Alonso del Arte_, Mar 18 2013

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# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/

Search: seq:1,1,2,3,5,8,13,21
Showing 11-20 of 92

%I A177246
%S A177246 1,1,2,3,5,8,13,21,55,89,233,377,610,987,1597,6765,17711,28657,75025,
%T A177246 121393,317811,2178309,5702887,39088169,1836311903,2971215073,
%U A177246 12586269025,32951280099,53316291173,86267571272,591286729879
%N A177246 Fibonacci numbers whose decimal expansion does not contain any digit 4.
%C A177246 Probability that Fib(n) contains no 4's goes to zero as n grows to infinity. I suppose that the maximum number is Fib(114) having 24 digits, none of them being a "4".
%H A177246 Robert Israel, <a href="/A177246/b177246.txt">Table of n, a(n) for n = 1..41</a> (conjectured to be complete)
%e A177246 a(9)=55 is the 9th Fibonacci having no digit 4's.
%p A177246 remove(t -> has(convert(t,base,10),4), map(combinat:-fibonacci, [$1..1000])); # _Robert Israel_, Dec 13 2018
%t A177246 Select[Fibonacci@Range@114, !MemberQ[IntegerDigits[#], 4] &] (* _Amiram Eldar_, Dec 13 2018 *)
%Y A177246 Cf. A000045, A177194, A177195, A177231, A177245, A176253.
%K A177246 nonn,base
%O A177246 1,3
%A A177246 _Carmine Suriano_, May 06 2010

%I A261575
%S A261575 0,1,1,2,3,5,8,13,21,34,55,29,1,24,2,53,3,17,6,10,10,27,16,37,26,4,43,
%T A261575 41,9,1,45,52,1,26,2,3,11,55,4,37,57,7,48,52,12,25,50,20,13,43,33,38,
%U A261575 33,54,51,16,28,1,29,50,22,2,20,7,51,3,49,57,13,6,9
%N A261575 Table of Fibonacci numbers in base 60 representation: row n contains the sexagesimal digits of A000045(n) in reversed order.
%C A261575 A261585(n) = length of n-th row;
%C A261575 T(n,0) = A261606(n) = in base 60: last sexagesimal digit of A000045(n);
%C A261575 T(n,A261607(n)-1) = A261607(n) = in base 60: initial  sexagesimal digit of A000045(n);
%C A261575 A000045(n) = sum(T(n,k)*60^k : k = 0..A261585(n)-1).
%H A261575 Reinhard Zumkeller, <a href="/A261575/b261575.txt">Rows n = 0..1000 of triangle, flattened</a>
%H A261575 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Sexagesimal.html">Sexagesimal</a>
%H A261575 Wikipedia, <a href="http://www.wikipedia.org/wiki/Sexagesimal">Sexagesimal</a>
%e A261575 A000045(42) = 20*60^4 + 40*60^3 + 20*60^2 + 38*60^1 + 16*60^0 = 267914296.
%e A261575 . ----------------------------------------------------------------------
%e A261575 .   n | T(n,*)       n | T(n,*)             n | T(n,*)
%e A261575 . ----+---------   ----+---------------   ----+-------------------------
%e A261575 .   0 | [0]         21 | [26,2,3]          42 | [16,38,20,40,20]
%e A261575 .   1 | [1]         22 | [11,55,4]         43 | [17,7,55,26,33]
%e A261575 .   2 | [1]         23 | [37,57,7]         44 | [33,45,15,7,54]
%e A261575 .   3 | [2]         24 | [48,52,12]        45 | [50,52,10,34,27,1]
%e A261575 .   4 | [3]         25 | [25,50,20]        46 | [23,38,26,41,21,2]
%e A261575 .   5 | [5]         26 | [13,43,33]        47 | [13,31,37,15,49,3]
%e A261575 .   6 | [8]         27 | [38,33,54]        48 | [36,9,4,57,10,6]
%e A261575 .   7 | [13]        28 | [51,16,28,1]      49 | [49,40,41,12,0,10]
%e A261575 .   8 | [21]        29 | [29,50,22,2]      50 | [25,50,45,9,11,16]
%e A261575 .   9 | [34]        30 | [20,7,51,3]       51 | [14,31,27,22,11,26]
%e A261575 .  10 | [55]        31 | [49,57,13,6]      52 | [39,21,13,32,22,42]
%e A261575 .  11 | [29,1]      32 | [9,5,5,10]        53 | [53,52,40,54,33,8,1]
%e A261575 .  12 | [24,2]      33 | [58,2,19,16]      54 | [32,14,54,26,56,50,1]
%e A261575 .  13 | [53,3]      34 | [7,8,24,26]       55 | [25,7,35,21,30,59,2]
%e A261575 .  14 | [17,6]      35 | [5,11,43,42]      56 | [57,21,29,48,26,50,4]
%e A261575 .  15 | [10,10]     36 | [12,19,7,9,1]     57 | [22,29,4,10,57,49,7]
%e A261575 .  16 | [27,16]     37 | [17,30,50,51,1]   58 | [19,51,33,58,23,40,12]
%e A261575 .  17 | [37,26]     38 | [29,49,57,0,3]    59 | [41,20,38,8,21,30,20]
%e A261575 .  18 | [4,43]      39 | [46,19,48,52,4]   60 | [0,12,12,7,45,10,33]
%e A261575 .  19 | [41,9,1]    40 | [15,9,46,53,7]    61 | [41,32,50,15,6,41,53]
%e A261575 .  20 | [45,52,1]   41 | [1,29,34,46,12]   62 | [41,44,2,23,51,51,26,1]
%o A261575 (Haskell)
%o A261575 a261575 n k = a261575_tabf !! n !! k
%o A261575 a261575_row n = a261575_tabf !! n
%o A261575 a261575_tabf = [0] : [1] :
%o A261575    zipWith (add 0) (tail a261575_tabf) a261575_tabf where
%o A261575    add c (a:as) (b:bs) = y : add c' as bs where (c', y) = divMod (a+b+c) 60
%o A261575    add c (a:as) [] = y : add c' as [] where (c', y) = divMod (a+c) 60
%o A261575    add 1 _ _ = [1]
%o A261575    add _ _ _ = []
%Y A261575 Cf. A000045, A261585 (row lengths), A261587 (row sums), A261598 (row products), A261606 (left edge), A261607 (right edge).
%K A261575 nonn,tabf,base
%O A261575 0,4
%A A261575 _Reinhard Zumkeller_, Sep 09 2015

%I A000044 M0691 N0255
%S A000044 1,1,1,2,3,5,8,13,21,34,55,89,144,232,375,606,979,1582,2556,4130,6673,
%T A000044 10782,17421,28148,45480,73484,118732,191841,309967,500829,809214,
%U A000044 1307487,2112571,3413385,5515174,8911138,14398164,23263822,37588502,60733592,98130253,158553878,256183302,413927966,668803781,1080619176,1746009572,2821113574,4558212008
%N A000044 Dying rabbits: a(0) = 1; for 1 <= n <= 12, a(n) = Fibonacci(n); for n >= 13, a(n) = a(n-1) + a(n-2) - a(n-13).
%C A000044 A107358 is a more satisfactory version, but I have left the present sequence unchanged (except for making the definition clearer) since it has been in the OEIS so long.
%C A000044 Number of compositions of n into parts 1, 3, 5, 7, 9, and 11. - _Joerg Arndt_, Sep 05 2014
%C A000044 If a(0) = 1 then it is not clear why a(2) = 1, it should be equal to a(1) + a(0) = 2. Does the first comment mean that a(0) is erroneous and should read a(0) = 0? In contrast to A107358, the term a(13) = 232 = 144 + 89 - 1 seems correct, since in this month the first and oldest pair of rabbits die. But a(14) should be equal to a(13) + a(12) = 232 + 144 because the first pair (which was also the only one present in month 2) has already died and there is no other pair aged 12 months. In general, the number of pairs which die in month n because they are aged exactly 12 months, equals a(n-14): this is the number of newborn pairs in month n - 12, viz. a(n-12) = a(n-13) [those from preceding month] + a(n-14) [the newborn ones] - #(those which die). - _M. F. Hasler_, Oct 06 2017
%D A000044 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000044 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000044 Harvey P. Dale, <a href="/A000044/b000044.txt">Table of n, a(n) for n = 0..1000</a>
%H A000044 J. H. E. Cohn, <a href="http://www.fq.math.ca/Scanned/2-2/cohn1.pdf">Letter to the editor</a>, Fib. Quart. 2 (1964), 108.
%H A000044 V. E. Hoggatt, Jr. and D. A. Lind, <a href="http://www.fq.math.ca/Scanned/7-5/hoggatt.pdf">The dying rabbit problem</a>, Fib. Quart. 7 (1969), 482-487.
%H A000044 <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1).
%F A000044 G.f.: 1/(1 - z - z^3 - z^5 - z^7 - z^9 -z^11).
%F A000044 G.f. A(x) = 1 / (1 - x / (1 - x^2 / (1 + x^10 / (1 + x^2 / (1 - x^2 / (1 + x^6 / (1 + x^2 / (1 - x^2 / (1 + x^2))))))))). - _Michael Somos_, Jan 04 2013
%F A000044 For n >= 11, a(n) = a(n-1) + a(n-3) + a(n-5) + a(n-7) + a(n-9) + a(n-11). - _Eric M. Schmidt_, Sep 04 2014
%e A000044 G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 8*x^6 + 13*x^7 + 21*x^8 + 34*x^9 + ...
%p A000044 with(combinat); f:=proc(n) option remember; if n=0 then RETURN(1); fi; if n <= 12 then RETURN(fibonacci(n)); fi; f(n-1)+f(n-2)-f(n-13); end;
%t A000044 CoefficientList[Series[1/(1 - z - z^3 - z^5 - z^7 - z^9 - z^11), {z, 0, 200}], z] (* _Vladimir Joseph Stephan Orlovsky_, Jun 10 2011 *)
%t A000044 LinearRecurrence[{1,1,0,0,0,0,0,0,0,0,0,0,-1},{1,1,1,2,3,5,8,13,21,34,55,89,144},100] (* _Harvey P. Dale_, Mar 24 2012 *)
%o A000044 (MAGMA) [ n eq 1 select 1 else n le 13 select Fibonacci(n-1) else Self(n-1)+Self(n-2)-Self(n-13): n in [1..50] ]; // _Klaus Brockhaus_, Dec 21 2010
%o A000044 (PARI) Vec(1/(1-z-z^3-z^5-z^7-z^9-z^11)+O(z^50)) \\ _Charles R Greathouse IV_, Jun 10 2011
%Y A000044 Cf. A107358. See A000045 for the Fibonacci numbers.
%K A000044 nonn,easy
%O A000044 0,4
%A A000044 _N. J. A. Sloane_; entry revised May 25 2005
%E A000044 G.f. corrected by _Charles R Greathouse IV_, Jun 10 2011

%I A048817
%S A048817 1,1,2,3,5,8,13,21,29,37,66,95,161,227,293,520,813,1106,1919,2732,
%T A048817 3545,4358,7903,12261,20164,28067,48231,68395,88559,156954,245513,
%U A048817 334072,579585,825098,1070611,1316124,1561637,2877761,4439398,6001035
%N A048817 Numerators of convergents to A058914.
%D A048817 S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 441-443.
%H A048817 Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/constant/cntfrc/mnkwsk.html">Minkowski's Question Mark Function</a> [Broken link]
%H A048817 Steven R. Finch, <a href="http://web.archive.org/web/20010208143502/http://www.mathsoft.com/asolve/constant/cntfrc/mnkwsk.html">Minkowski's Question Mark Function</a> [From the Wayback machine]
%H A048817 <a href="/index/Me#MinkowskiQ">Index entries for Minkowski's question mark function</a>
%H A048817 <a href="/index/Me#MinkowskiQ">Index entries for sequences related to Minkowski's question mark function</a>
%e A048817 f(0)=0, f(1/8)=1/4, f(1/4)=1/3, f(3/8)=2/5, f(1/2)=1/2, f(5/8)=3/5...
%e A048817 1/2, 1/3, 2/5, 3/7, 5/12, 8/19, ...
%Y A048817 Denominators in A048818. Cf. A048819-A048822.
%Y A048817 Associated continued fraction in A058914. Denominators in A048818.
%K A048817 nonn,frac
%O A048817 0,3
%A A048817 _Christian G. Bower_, Apr 15 1999

%I A105471
%S A105471 0,1,1,2,3,5,8,13,21,34,55,89,44,33,77,10,87,97,84,81,65,46,11,57,68,
%T A105471 25,93,18,11,29,40,69,9,78,87,65,52,17,69,86,55,41,96,37,33,70,3,73,
%U A105471 76,49,25,74,99,73,72,45,17,62,79,41,20,61,81,42,23,65,88,53,41,94,35,29,64
%N A105471 a(n) = Fibonacci(n) mod 100.
%C A105471 a(n) = A105472(n)*10 + A003893(n);
%C A105471 the sequence is periodic with period 300; all blocks of 60 successive terms contain 20 even and 40 odd numbers, see A003893.
%H A105471 Vincenzo Librandi, <a href="/A105471/b105471.txt">Table of n, a(n) for n = 0..1000</a>
%H A105471 <a href="/index/Rec#order_300">Index entries for linear recurrences with constant coefficients</a>, order 300.
%F A105471 a(n) = (a(n-1) + a(n-2)) mod 100 for n>1, a(0) = 0, a(1) = 1.
%t A105471 Mod[Fibonacci[Range[0,100]],100] (* _Harvey P. Dale_, Jun 12 2014 *)
%o A105471 (MAGMA) [Fibonacci(n) mod(100): n in [0..75]]; // _Vincenzo Librandi_, Jul 23 2014
%o A105471 (Haskell)
%o A105471 a105471 n = a105471_list !! n
%o A105471 a105471_list = 0 : 1 :
%o A105471    zipWith ((flip mod 100 .) . (+)) a105471_list (tail a105471_list)
%o A105471 -- _Reinhard Zumkeller_, Aug 06 2014
%o A105471 (PARI) a(n)=fibonacci(n%300)%100 \\ _Charles R Greathouse IV_, Oct 07 2015
%Y A105471 Cf. A000045.
%K A105471 nonn,easy
%O A105471 0,4
%A A105471 _Reinhard Zumkeller_, Apr 09 2005

%I A243063
%S A243063 1,1,2,3,5,8,13,21,34,55,89,144,233,377,61,438,499,937,1436,2373,389,
%T A243063 2762,3151,5913,964,6877,7841,14718,22559,37277,59836,97113,156949,
%U A243063 25462,182411,27873,21284,49157,7441,56598,6439,6337,12776,19113,31889,512,3241
%N A243063 Numbers generated by a Fibonacci-like sequence in which zeros are suppressed.
%C A243063 Let x(1) = 1, x(2) = 1, then begin the sequence x(i) = no-zero(x(i-2) + x(i-1)), where the function no-zero(n) removes all zero digits from n.
%C A243063 The sequence behaves like a standard Fibonacci sequence until step 15, where x = no-zero(233 + 377) = no-zero(610) = 61. At step 16, x = 377 + 61 = 438. The sequence then proceeds until step 927, where x = no-zero(224 + 377) = no-zero(601) = 61. Therefore at step 928, x = 377 + 61 = 438 and the sequence repeats.
%H A243063 Anthony Sand, <a href="/A243063/b243063.txt">Table of n, a(n) for n = 1..927</a>
%F A243063 x(i) = no-zero(x(i-2) + x(i-1)). For example, no-zero(233 + 377) = no-zero(610) = 61.
%e A243063 x(3) = x(1) + x(2) = 1 + 1 = 2.
%e A243063 x(4) = x(2) + x(3) = 1 + 2 = 3.
%e A243063 x(15) = no-zero(x(13) + x(14)) = no-zero(233 + 377) = no-zero(610) = 61.
%e A243063 x(16) = 377 + 61 = 438.
%p A243063 noz:=proc(n) local a,t1,i,j; a:=0; t1:=convert(n,base,10); for i from 1 to nops(t1) do j:=t1[nops(t1)+1-i]; if j <> 0 then a := 10*a+j; fi; od: a; end; # A004719
%p A243063 t1:=[1,1]; for n from 3 to 100 do t1:=[op(t1),noz(t1[n-1]+t1[n-2])]; od: t1; # _N. J. A. Sloane_, Jun 11 2014
%Y A243063 Cf. A000045, A004719, A242350, A243657, A243658, A306773.
%K A243063 nonn,base
%O A243063 1,3
%A A243063 _Anthony Sand_, Jun 09 2014

%I A185357
%S A185357 1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4180,6763,
%T A185357 10942,17703,28642,46340,74974,121301,196254,317521,513720,831152,
%U A185357 1344728,2175647,3519998,5695035,9214046,14907484,24118947,39022252,63134437,102145749
%N A185357 Expansion of 1/(1 - x - x^2 + x^18 - x^20).
%C A185357 Limiting ratio is 1.61791..., the real root of -1 + x^2 - x^18 - x^19 + x^20. Signature in Mathematica is:
%C A185357 -CoefficientList[1 - x - x^2 + x^18 - x^20, x]
%C A185357 {-1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1}.
%C A185357 The sequence agrees with the Fibonacci numbers (A000045) for the first 18 terms.
%H A185357 G. C. Greubel, <a href="/A185357/b185357.txt">Table of n, a(n) for n = 0..1000</a>
%H A185357 David Terr and Eric W. Weisstein, <a href="http://mathworld.wolfram.com/PisotNumber.html">MathWorld: Pisot Number</a>
%H A185357 <a href="/index/Rec#order_20">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1).
%t A185357 CoefficientList[Series[1/(1 - x - x^2 + x^18 - x^20), {x, 0, 50}], x]
%o A185357 (PARI) Vec(1/(1-x-x^2+x^18-x^20) + O(x^50)) \\ _G. C. Greubel_, Nov 16 2016
%Y A185357 Cf. A117791, A107293, A204631.
%K A185357 nonn,easy
%O A185357 0,3
%A A185357 _Roger L. Bagula_, Jan 21 2012

%I A204026
%S A204026 1,1,1,1,2,1,1,2,2,1,1,2,3,2,1,1,2,3,3,2,1,1,2,3,5,3,2,1,1,2,3,5,5,3,
%T A204026 2,1,1,2,3,5,8,5,3,2,1,1,2,3,5,8,8,5,3,2,1,1,2,3,5,8,13,8,5,3,2,1,1,2,
%U A204026 3,5,8,13,13,8,5,3,2,1,1,2,3,5,8,13,21,13,8,5,3,2,1,1,2,3,5,8
%N A204026 Symmetric matrix based on f(i,j)=min(F(i+1),F(j+1)), where F=A000045 (Fibonacci numbers), by antidiagonals.
%C A204026 A204026 represents the matrix M given by f(i,j)=min(F(i+1),F(j+1)) for i>=1 and j>=1.  See A204027 for characteristic polynomials of principal submatrices of M, with interlacing zeros.  See A204016 for a guide to other choices of M.
%e A204026 Northwest corner:
%e A204026 1 1 1 1 1 1
%e A204026 1 2 2 2 2 2
%e A204026 1 2 3 3 3 3
%e A204026 1 2 3 5 5 5
%e A204026 1 2 3 5 8 8
%e A204026 1 2 3 5 8 13
%t A204026 f[i_, j_] := Min[Fibonacci[i + 1], Fibonacci[j + 1]]
%t A204026 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A204026 TableForm[m[6]] (* 6x6 principal submatrix *)
%t A204026 Flatten[Table[f[i, n + 1 - i],
%t A204026   {n, 1, 15}, {i, 1, n}]]  (* A204026 *)
%t A204026 p[n_] := CharacteristicPolynomial[m[n], x];
%t A204026 c[n_] := CoefficientList[p[n], x]
%t A204026 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A204026 Table[c[n], {n, 1, 12}]
%t A204026 Flatten[%]                 (* A204027 *)
%t A204026 TableForm[Table[c[n], {n, 1, 10}]]
%Y A204026 Cf. A204026, A204016, A202453.
%K A204026 nonn,tabl
%O A204026 1,5
%A A204026 _Clark Kimberling_, Jan 11 2012

%I A226999
%S A226999 1,0,1,1,2,3,5,8,13,21,35,55,93,149,248,403,671,1098,1827,3013,5013,
%T A226999 8313,13859,23063,38534,64341,107715,180355,302565,507784,853507,
%U A226999 1435415,2416941,4072272,6868062,11590807,19577555,33088481,55964327,94712212
%N A226999 Inverse Euler transform of A005169 (fountains of coins).
%C A226999 If G005169(x) = Sum_{i>=0} A005169(n)*x^n is the generating function of A005169, the a(n) are defined through G005169(x) = Product_{n>=1} 1/(1-x^n)^a(n), the inverse Euler transform of A005169.
%H A226999 Alois P. Heinz, <a href="/A226999/b226999.txt">Table of n, a(n) for n = 1..4178</a>
%H A226999 R. K. Guy, <a href="/A005169/a005169_6.pdf">Letter to N. J. A. Sloane</a>, Sep 25 1986.
%H A226999 R. K. Guy, <a href="/A005728/a005728.pdf">Letter to N. J. A. Sloane, 1987</a>
%H A226999 R. K. Guy, <a href="http://www.jstor.org/stable/2322249">The strong law of small numbers</a>. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
%H A226999 R. K. Guy, <a href="/A005165/a005165.pdf">The strong law of small numbers</a>. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
%Y A226999 Cf. A005170.
%K A226999 nonn
%O A226999 1,5
%A A226999 _R. J. Mathar_, Jun 26 2013

%I A261587
%S A261587 0,1,1,2,3,5,8,13,21,34,55,30,26,56,23,20,43,63,47,51,98,31,70,101,
%T A261587 112,95,89,125,96,103,81,125,29,95,65,101,48,149,138,169,130,122,134,
%U A261587 138,154,174,151,148,122,152,156,131,169,241,233,179,235,178,236
%N A261587 Sum of sexagesimal digits of Fibonacci numbers in base 60 representation.
%C A261587 a(n) = sum of n-th row in table A261575.
%H A261587 Reinhard Zumkeller, <a href="/A261587/b261587.txt">Table of n, a(n) for n = 0..10000</a>
%H A261587 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Sexagesimal.html">Sexagesimal</a>
%H A261587 Wikipedia, <a href="http://www.wikipedia.org/wiki/Sexagesimal">Sexagesimal</a>
%o A261587 (Haskell)
%o A261587 a261587 = sum . a261575_row
%Y A261587 Cf. A261575, A000045, A004090, A261598.
%K A261587 nonn,base
%O A261587 0,4
%A A261587 _Reinhard Zumkeller_, Sep 09 2015

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# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/

Search: seq:1,1,2,3,5,8,13,21
Showing 21-30 of 92

%I A261598
%S A261598 0,1,1,2,3,5,8,13,21,34,55,29,48,159,102,100,432,962,172,369,2340,156,
%T A261598 2420,14763,29952,25000,18447,67716,22848,63800,21420,217854,2250,
%U A261598 35264,34944,99330,14364,1300500,0,8726016,2303910,544272,9728000,5615610,8419950
%N A261598 Product of sexagesimal digits of Fibonacci numbers in base 60 representation.
%C A261598 a(n) = product of n-th row in table A261575;
%C A261598 conjecture: a(n) = 0 for n > 3329 (empirically checked up to 36000).
%H A261598 Reinhard Zumkeller, <a href="/A261598/b261598.txt">Table of n, a(n) for n = 0..10000</a>
%H A261598  Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Sexagesimal.html">Sexagesimal</a>
%H A261598 Wikipedia, <a href="http://www.wikipedia.org/wiki/Sexagesimal">Sexagesimal</a>
%o A261598 (Haskell)
%o A261598 a261598 = product . a261575_row
%Y A261598  Cf. A261575, A000045, A246558, A261587.
%K A261598 nonn,base
%O A261598 0,4
%A A261598 _Reinhard Zumkeller_, Sep 09 2015

%I A261606
%S A261606 0,1,1,2,3,5,8,13,21,34,55,29,24,53,17,10,27,37,4,41,45,26,11,37,48,
%T A261606 25,13,38,51,29,20,49,9,58,7,5,12,17,29,46,15,1,16,17,33,50,23,13,36,
%U A261606 49,25,14,39,53,32,25,57,22,19,41,0,41,41,22,3,25,28,53
%N A261606 a(n) = Fibonacci(n) mod 60.
%C A261606 a(n) = A261575(n,0).
%C A261606 Periodic with period length 120. - _Ray Chandler_, Sep 23 2015
%H A261606 Reinhard Zumkeller, <a href="/A261606/b261606.txt">Table of n, a(n) for n = 0..10000</a>
%H A261606 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Sexagesimal.html">Sexagesimal</a>
%H A261606 Wikipedia, <a href="http://www.wikipedia.org/wiki/Sexagesimal">Sexagesimal</a>
%F A261606 a(0) = 0, a(1) = 1 and for n > 1: a(n) = (a(n-1) + a(n-2)) mod 60.
%o A261606 (Haskell)
%o A261606 a261606 n = a261606_list !! n
%o A261606 a261606_list = 0 : 1 : map (flip mod 60)
%o A261606                            (zipWith (+) a261606_list $ tail a261606_list)
%Y A261606 Cf. A000045, A261575, A261607, A003893.
%K A261606 nonn,base
%O A261606 0,4
%A A261606 _Reinhard Zumkeller_, Sep 09 2015

%I A261607
%S A261607 0,1,1,2,3,5,8,13,21,34,55,1,2,3,6,10,16,26,43,1,1,3,4,7,12,20,33,54,
%T A261607 1,2,3,6,10,16,26,42,1,1,3,4,7,12,20,33,54,1,2,3,6,10,16,26,42,1,1,2,
%U A261607 4,7,12,20,33,53,1,2,3,6,9,16,25,42,1,1,2,4,7,12
%N A261607 Initial digit of Fibonacci number F(n) in base 60.
%C A261607 a(n) = A261575(n,A261585(n)-1).
%H A261607 Reinhard Zumkeller, <a href="/A261607/b261607.txt">Table of n, a(n) for n = 0..1000</a>
%H A261607 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Sexagesimal.html">Sexagesimal</a>
%H A261607 Wikipedia, <a href="http://www.wikipedia.org/wiki/Sexagesimal">Sexagesimal</a>
%o A261607 (Haskell)
%o A261607 a261607 = last . a261575_row
%Y A261607 Cf. A000045, A261575, A261585, A261606, A008963.
%K A261607 nonn,base
%O A261607 0,4
%A A261607 _Reinhard Zumkeller_, Sep 09 2015

%I A023438
%S A023438 0,1,1,2,3,5,8,13,21,33,53,84,134,213,339,539,857,1363,2167,3446,5479,
%T A023438 8712,13852,22025,35020,55682,88535,140771,223827,355886,565861,
%U A023438 899722,1430563,2274603,3616631,5750463
%N A023438 Dying rabbits: a(n) = a(n-1) + a(n-2) - a(n-8).
%H A023438 J. H. E. Cohn, <a href="http://www.fq.math.ca/Scanned/2-2/cohn1.pdf">Letter to the editor</a>, Fib. Quart. 2 (1964), 108.
%H A023438 V. E. Hoggatt, Jr. and D. A. Lind, <a href="http://www.fq.math.ca/Scanned/7-5/hoggatt.pdf">The dying rabbit problem</a>, Fib. Quart. 7 (1969), 482-487.
%H A023438 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,0,0,0,0,-1).
%F A023438 G.f.: x / ( (x-1)*(x^7+x^6+x^5+x^4+x^3+x^2-1) ). - _R. J. Mathar_, Nov 29 2011
%o A023438 (PARI) concat(0, Vec(x / ( (x-1)*(x^7+x^6+x^5+x^4+x^3+x^2-1) ) + O(x^60))) \\ _Michel Marcus_, Sep 06 2017
%Y A023438 See A000045 for the Fibonacci numbers.
%K A023438 nonn
%O A023438 0,4
%A A023438 _N. J. A. Sloane_.

%I A077371
%S A077371 0,1,1,2,3,5,8,13,21,34,55,89,233,610,987
%N A077371 Fibonacci numbers whose internal digits form a Fibonacci number. Equivalently, Fibonacci numbers from which deleting the MSD and LSD leaves a Fibonacci number.
%C A077371 Conjecture: The sequence is finite.
%C A077371 No more terms < 10^6. - _Lars Blomberg_, May 20 2015
%C A077371 From _Manfred Scheucher_, Jun 02 2015 (Start)
%C A077371 No more terms < 10^10000.
%C A077371 When considering binary representations, the sequence would be 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 144, and no further terms < 2^150000 (about 10^44095).
%C A077371 When considering k-ary representations with k=2..100, each of the sequences has some small terms in the beginning (as in the 10-ary case) and no further terms <10^1000.
%C A077371 The sequence seems to be finite for any base, not just for base 10.
%C A077371 Another observation: When considering k-ary representations with k=55,144,377,... (Fibonacci numbers with even index, A001906), the number of "initial terms" (terms <10^1000) increases very fast.
%C A077371 (End)
%H A077371 Manfred Scheucher, <a href="/A077371/a077371.sage.txt">Sage Script</a>
%Y A077371 Cf. A077372, A077373, A077374, A077375.
%K A077371 base,more,nonn
%O A077371 1,4
%A A077371 _Amarnath Murthy_, Nov 06 2002

%I A077372
%S A077372 0,1,1,2,3,5,8,13,21,34,55,89,121393,1836311903,2504730781961,
%T A077372 10610209857723,10284720757613717413913,184551825793033096366333,
%U A077372 59425114757512643212875125,155576970220531065681649693
%N A077372 Fibonacci numbers whose external digits form a Fibonacci number. Or Fibonacci numbers whose MSD and LSD form a Fibonacci number.
%C A077372 Sequence can be easily generated up to >10000 terms with the appended Sage script. - _Manfred Scheucher_, Jun 02 2015
%H A077372 Manfred Scheucher, <a href="/A077372/b077372.txt">Table of n, a(n) for n = 1..100</a>
%H A077372 Manfred Scheucher, <a href="/A077372/a077372.sage.txt">Sage Script</a>
%o A077372 (Sage) F=[fibonacci(i) for i in [0..200]]
%o A077372 [x for x in F if x<10 or (x>10 and (10*x.digits()[-1]+x.digits()[0]) in F)]
%o A077372 # _Tom Edgar_, Jun 03 2015
%Y A077372 Cf. A077371, A077373.
%K A077372 base,nonn
%O A077372 1,4
%A A077372 _Amarnath Murthy_, Nov 06 2002
%E A077372 More terms from _Lior Manor_, Nov 06 2002

%I A093089
%S A093089 1,1,2,3,5,8,13,21,34,55,89,1,44,90,45,1,34,1,35,46,35,35,36,81,81,70,
%T A093089 71,1,17,1,62,1,51,1,41,72,18,18,63,63,52,52,42,1,13,90,36,81,1,26,1,
%U A093089 15,1,4,94,43,14,1,3,1,26,1,17,82,27,27,16,16,5,98,1,37,57,15,4,4,27,27,18
%N A093089 "Fibonacci in pairs": start with a(1)=1, a(2)=1; repeatedly adjoin sum of previous two terms but chopped from the right into pairs of 2 digits.
%C A093089 Do all pairs of digits appear infinitely often? The sequence is not periodic.
%e A093089 ... a(11)=a(9)+a(10), a(12)=left pair of (a(10)+a(11)=55+89=1 44), a(13)=right pair of (a(10)+a(11)=55+89=1 44), a(14)=a(11)+a(12) ...
%Y A093089 Cf. A093086, A093087, A093088.
%K A093089 nonn,base
%O A093089 1,3
%A A093089 _Bodo Zinser_, Mar 20 2004

%I A132916
%S A132916 0,1,1,1,1,1,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,
%T A132916 4181,6765,10946,21892,39603,72441,133936,245980,452357,832273,
%U A132916 1530610,2815240,5178123,9523973,17517336,32219432,59260741,108997509,200477682
%N A132916 a(0)=0; a(1)=1; a(n) = Sum_{k=1..floor(n^(1/3))} a(n-k) for n >= 2.
%C A132916 Lim_{n->infinity} a(n+1)/a(n) = 2. Contrast with Fibonacci sequence. Also a(n+1)/a(n) = 2 iff n+1 >= 8 is a cube.
%C A132916 Up to a(26) = 10946, but not beyond, the sequence consists of the Fibonacci numbers A000045(0..21). - _M. F. Hasler_, May 10 2017
%H A132916 Robert Israel, <a href="/A132916/b132916.txt">Table of n, a(n) for n = 0..3345</a>
%F A132916 a(n) = Sum_{k=1..floor(n^(1/3))} a(n-k) for n >= 2; a(0)=0; a(1)=1.
%e A132916 a(27) = a(24) + a(25) + a(26) = 4181 + 6765 + 10946 = 21892.
%p A132916 f:= proc(n) option remember;
%p A132916 add(procname(n-k),k=1..floor(n^(1/3)))
%p A132916 end proc:
%p A132916 f(0):= 0: f(1):= 1:
%p A132916 map(f, [$0..50]); # _Robert Israel_, Dec 16 2018
%t A132916 a[n_] := a[n] = If[n < 2, n, Sum[a[n - k], {k, Floor[n^(1/3)]}]]; Array[a, 43, 0] (* _Michael De Vlieger_, May 10 2017 *)
%Y A132916 Cf. A000045, A132915.
%K A132916 nonn
%O A132916 0,9
%A A132916 _Rick L. Shepherd_, Sep 04 2007
%E A132916 Incorrect g.f. and programs deleted by _Colin Barker_, Dec 17 2018

%I A141169
%S A141169 0,0,1,0,1,1,0,1,1,2,0,1,1,2,3,0,1,1,2,3,5,0,1,1,2,3,5,8,0,1,1,2,3,5,
%T A141169 8,13,0,1,1,2,3,5,8,13,21,0,1,1,2,3,5,8,13,21,34,0,1,1,2,3,5,8,13,21,
%U A141169 34,55,0,1,1,2,3,5,8,13,21,34,55,89,0,1,1,2,3,5,8,13,21,34,55,89,144,0,1,1,2,3,5,8,13,21
%N A141169 Triangle of Fibonacci numbers, read by rows: T(n,k) = A000045(k), 0<=k<=n.
%C A141169 T(n,0) = A000004(n); T(n,n) = A000045(n);
%C A141169 central terms: T(2*n,n) = A000045(n);
%C A141169 sums of rows: Sum(T(n,k): 0<=k<=n) = A000071(n+2);
%C A141169 alternating sums of rows: Sum(T(n,k)*(-1)^k: 0<=k<=n) = A119282(n);
%C A141169 T(n,k) + T(n,n-k) = A094570(n,k).
%H A141169 Reinhard Zumkeller, <a href="/A141169/b141169.txt">Rows n=0..125 of triangle, flattened</a>
%o A141169 (Haskell)
%o A141169 import Data.List (inits)
%o A141169 a141169 n k = a141169_tabl !! n !! k
%o A141169 a141169_row n = a141169_tabl !! n
%o A141169 a141169_tabl = tail $ inits a000045_list
%o A141169 a141169_list = concat $ a141169_tabl
%o A141169 -- _Reinhard Zumkeller_, Aug 24 2015, Mar 21 2011
%Y A141169 Cf. A000045, A000004, A000071, A094570, A119282.
%K A141169 nonn,tabl
%O A141169 0,10
%A A141169 _Reinhard Zumkeller_, Mar 21 2011

%I A177247
%S A177247 1,1,2,3,5,8,13,21,34,55,89,144,233,377,987,1597,2584,4181,17711,
%T A177247 75025,121393,317811,514229,832040,2178309,3524578,5702887,14930352,
%U A177247 24157817,102334155,433494437,701408733,1134903170,2971215073,7778742049
%N A177247 Fibonacci numbers Fib(n) whose decimal expansion does not contain any digit 6.
%C A177247 Probability that Fib(n) contains no 6's goes to zero as n grows to infinity. I suppose the maximum number is F(258) having 54 digits with no 6's.
%e A177247 a(7)=13 since it is the 7th Fibonacci having no 6's
%Y A177247 Cf. A000045, A177194, A177195, A177231, A177245, A177246, A176253
%K A177247 nonn,base
%O A177247 1,3
%A A177247 _Carmine Suriano_, May 06 2010

# Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE

# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/

Search: seq:1,1,2,3,5,8,13,21
Showing 31-40 of 92

%I A229194
%S A229194 1,1,1,2,3,5,8,13,21,35,58,97,163,275,466,793,1353,2315,3969,6817,
%T A229194 11726,20195,34816,60073,103724,179195,309724,535537,926275,1602515,
%U A229194 2773034,4799353,8307516,14381675,24899377,43112257,74651790,129271235,223862687,387682633,671402698,1162785755,2013837368,3487832977,6040770648,10462450355,18120829034,31385253913,54359521280,94151567435,163072632198
%N A229194 Integers nearest to (2^((n-3)/2) + 3^((n-3)/2)).
%C A229194 This sequence illustrates the second law of small numbers because it is a coincidence that the terms for 1 <= n <= 8 are the same as the Fibonacci numbers F(n) (A000045): a(n) = F(n) for 1 <= n <= 8.
%C A229194 Furthermore, the following terms are the sum of two Fibonacci numbers: a(9) = F(9) + F(2), a(10) = F(10) + F(4), a(11) = F(11) + F(6), a(14) = F(14) + F(11); or the algebraic sum of three Fibonacci numbers: a(12) = F(12) + F(8) - F(3), a(13) = F(13) + F(10) - F(7), a(14) = F(14) + F(12) - F(10), a(18) = F(19) - F(13) - F(8), a(19) = F(20) + F(10) - F(4); or the algebraic sum of four Fibonacci numbers: a(15) = F(15) + F(12) + F(9) + F(5), a(16) = F(16) + F(14) - F(6) - F(4), a(17) = F(18) - F(13) - F(9) - F(3), a(18) = F(18) + F(16) + F(14) + F(8), a(19) = F(19) + F(18) + F(10) - F(3).
%C A229194 Note that, for following values of n, a(n) > F(n+1) for n >= 20.
%C A229194 Remark as well that (2^(1/2) + 3^(1/2)) = 3.14626437... ~=  Pi (see A135611).
%D A229194 T. Koshy, Fibonacci and Lucas Numbers with Applications, New York, Wiley-Interscience, 2001
%D A229194 I. Stewart, L'univers des nombres, Belin-Pour La Science, Paris 2000.
%H A229194 Vladimir Pletser, <a href="/A229194/b229194.txt">Table of n, a(n) for n = 0..1000</a>
%H A229194 R. K. Guy, <a href="http://www.jstor.org/stable/2691503">The Second Strong Law of Small Numbers</a>, Math. Mag, 63 (1990), no. 1, 3-20.
%H A229194 I. Stewart, <a href="http://www.whydomath.org/Reading_Room_Material/ian_stewart/9505.html">Fibonacci Forgeries</a>
%H A229194 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number.</a>
%H A229194 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StrongLawofSmallNumbers.html">Strong Law of Small Numbers.</a>
%F A229194 a(n) = round(2^((n-3)/2) + 3^((n-3)/2)).
%p A229194 seq(round(2^((n-3)/2)+3^((n-3)/2)), n=0..50);
%t A229194 Table[Round[2^((n - 3)/2) + 3^((n - 3)/2)], {n, 0, 50}] (* _Vincenzo Librandi_, Sep 20 2013 *)
%o A229194 (MAGMA) [Round(2^((n-3)/2) + 3^((n-3)/2)): n in [0..50]]; // _Vincenzo Librandi_, Sep 20 2013
%Y A229194 Cf. A000045, A005181.
%K A229194 nonn
%O A229194 0,4
%A A229194 _Vladimir Pletser_, Sep 15 2013

%I A232666
%S A232666 0,1,1,2,3,5,8,13,21,34,55,89,4,93,97,190,287,477,764,1241,2005,541,
%T A232666 2546,3087,5633,8720,14353,23073,37426,60499,97925,26404,124329,
%U A232666 150733,275062,425795,700857,1126652,1827509,2954161,796945,3751106,4548051,8299157,12847208,21146365,33993573
%N A232666 6-free Fibonacci numbers.
%C A232666 The sequences of n-free Fibonacci numbers were suggested by John H. Conway.
%C A232666 a(n) is the sum of the two previous terms divided by the largest possible power of 6.
%C A232666 4-free Fibonacci numbers are A224382.
%C A232666 The sequence coincides with the Fibonacci sequence until the first multiple of 6 in the Fibonacci sequence: 144, which in this sequence is divided by 36 to produce 4.
%C A232666 7-free Fibonacci numbers is A078414.
%H A232666 B. Avila, T. Khovanova, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Avila/avila4.html">Free Fibonacci Sequences</a>, J. Int. Seq. 17 (2014) # 14.8.5.
%t A232666 sixPower[n_] := (a = Transpose[FactorInteger[n]]; a2 = Position[a[[1]], 2]; a3 = Position[a[[1]], 3]; If[Length[a2] == 0 || Length[a3] == 0 , res = 0, res = Min[a[[2]][[a2[[1]][[1]]]], a[[2]][[a3[[1]][[1]]]]]]; res); sixFree[n_] := n/6^sixPower[n]; appendNext6Free[list_] := Append[list, sixFree[list[[-1]] + list[[-2]]]]; Nest[appendNext6Free, {0, 1}, 50]
%Y A232666 Cf. A224382, A214684.
%K A232666 nonn
%O A232666 0,4
%A A232666 _Brandon Avila_ and _Tanya Khovanova_, Nov 27 2013

%I A321021
%S A321021 0,1,1,2,3,5,8,13,21,34,0,34,34,68,102,170,7,1,8,9,17,26,43,69,2,71,
%T A321021 73,1,74,75,149,4,153,157,310,467,0,467,467,934,40,974,4,978,982,1960,
%U A321021 94,2054,2148,40,21,61,82,143,5,148,153,301,5,306,3
%N A321021 a(0)=0, a(1)=1; thereafter a(n) = a(n-2)+a(n-1), keeping just the digits that appear exactly once.
%C A321021 a(n) = A320486(a(n-2)+a(n-1)).
%C A321021 This must eventually enter a cycle, since there are only finitely many pairs of numbers that both have distinct digits. In fact, at step 171, enters a cycle of length 100 (see A321022).
%C A321021 Another entry into this cycle would be to start with 2, 1 and use the same rule, in which case the sequence would begin (2, 1, 3, 4, 7, 0, 7, 7, 14, 21, 35, 56, 91, 147, 238, 385, 623, ..., 40, 80, 120), a cycle of length 100 that repeats (cf. A321022).
%H A321021 N. J. A. Sloane, <a href="/A321021/b321021.txt">Table of n, a(n) for n = 0..1000</a>
%p A321021 f:= proc(n) local F, S;
%p A321021   F:= convert(n, base, 10);
%p A321021   S:= select(t -> numboccur(t, F)>1, [$0..9]);
%p A321021   if S = {} then return n fi;
%p A321021   F:= subs(seq(s=NULL, s=S), F);
%p A321021   add(F[i]*10^(i-1), i=1..nops(F))
%p A321021 end proc: # A320486
%p A321021 x:=0: y:=1: lprint(x); lprint(y);
%p A321021 for n from 2 to 500 do
%p A321021 z:=f(x+y); lprint(z); x:=y; y:=z; od:
%Y A321021 Cf. A000045 (Fibonacci), A320486 (Angelini's contraction), A321022.
%K A321021 nonn,base
%O A321021 0,4
%A A321021 _N. J. A. Sloane_, Nov 19 2018

%I A023439
%S A023439 0,1,1,2,3,5,8,13,21,34,54,87,139,223,357,572,916,1467,2349,3762,6024,
%T A023439 9647,15448,24738,39614,63436,101583,162670,260491,417137,667981,
%U A023439 1069670,1712913,2742969,4392446,7033832
%N A023439 Dying rabbits: a(n) = a(n-1) + a(n-2) - a(n-9).
%H A023439 Floris P. van Doorn and Jasper Mulder, <a href="/A023439/b023439.txt">Table of n, a(n) for n = 0..2000</a>
%H A023439 J. H. E. Cohn, <a href="http://www.fq.math.ca/Scanned/2-2/cohn1.pdf">Letter to the editor</a>, Fib. Quart. 2 (1964), 108.
%H A023439 V. E. Hoggatt, Jr. and D. A. Lind, <a href="http://www.fq.math.ca/Scanned/7-5/hoggatt.pdf">The dying rabbit problem</a>, Fib. Quart. 7 (1969), 482-487.
%H A023439 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,0,0,0,0,0,-1).
%F A023439 G.f.: x / ( (x-1)*(1+x)*(x^7+x^5+x^3+x-1) ). - _R. J. Mathar_, Nov 29 2011
%Y A023439 See A000045 for the Fibonacci numbers.
%K A023439 nonn
%O A023439 0,4
%A A023439 _N. J. A. Sloane_

%I A023440
%S A023440 0,1,1,2,3,5,8,13,21,34,55,88,142,228,367,590,949,1526,2454,3946,6345,
%T A023440 10203,16406,26381,42420,68211,109682,176367,283595,456016,733266,
%U A023440 1179079,1895939,3048637,4902156,7882582
%N A023440 Dying rabbits: a(n) = a(n-1) + a(n-2) - a(n-10).
%H A023440 Matthew House, <a href="/A023440/b023440.txt">Table of n, a(n) for n = 0..4825</a>
%H A023440 John H. E. Cohn, <a href="http://www.fq.math.ca/Scanned/2-2/cohn1.pdf">Letter to the editor</a>, Fib. Quart. 2 (1964), 108.
%H A023440 V. E. Hoggatt, Jr. and D. A. Lind, <a href="http://www.fq.math.ca/Scanned/7-5/hoggatt.pdf">The dying rabbit problem</a>, Fib. Quart. 7 (1969), 482-487.
%H A023440 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,0,0,0,0,0,0,-1).
%F A023440 G.f.: x/((x - 1)*(x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 - 1)). - _R. J. Mathar_, Nov 29 2011
%Y A023440 See A000045 for the Fibonacci numbers.
%K A023440 nonn,easy
%O A023440 0,4
%A A023440 _N. J. A. Sloane_

%I A041247
%S A041247 1,1,2,3,5,8,13,21,475,496,971,1467,2438,3905,6343,10248,231799,
%T A041247 242047,473846,715893,1189739,1905632,3095371,5001003,113117437,
%U A041247 118118440,231235877,349354317,580590194,929944511,1510534705,2440479216,55201077457,57641556673
%N A041247 Denominators of continued fraction convergents to sqrt(135).
%H A041247 Vincenzo Librandi, <a href="/A041247/b041247.txt">Table of n, a(n) for n = 0..200</a>
%H A041247 <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,488,0,0,0,0,0,0,0,-1).
%F A041247 G.f.: -(x^2-x-1)*(x^4+3*x^2+1)*(x^8+7*x^4+1) / (x^16-488*x^8+1). - _Colin Barker_, Nov 14 2013
%F A041247 a(n) = 488*a(n-8) - a(n-16). - _Vincenzo Librandi_, Dec 13 2013
%t A041247 Denominator[Convergents[Sqrt[135], 30]]  (* _Vincenzo Librandi_, Dec 13 2013 *)
%o A041247 (MAGMA) I:=[1,1,2,3,5,8,13,21,475,496,971,1467,2438,3905,6343,10248]; [n le 16 select I[n] else 488*Self(n-8)-Self(n-16): n in [1..40]]; // _Vincenzo Librandi_, Dec 13 2013
%Y A041247 Cf. A041246, A010194.
%K A041247 nonn,frac,easy
%O A041247 0,3
%A A041247 _N. J. A. Sloane_.
%E A041247 More terms from _Colin Barker_, Nov 14 2013

%I A042581
%S A042581 1,1,2,3,5,8,13,21,34,55,3114,3169,6283,9452,15735,25187,40922,66109,
%T A042581 107031,173140,9802871,9976011,19778882,29754893,49533775,79288668,
%U A042581 128822443,208111111,336933554,545044665,30859434794,31404479459,62263914253,93668393712
%N A042581 Denominators of continued fraction convergents to sqrt(819).
%H A042581 Vincenzo Librandi, <a href="/A042581/b042581.txt">Table of n, a(n) for n = 0..200</a>
%H A042581 <a href="/index/Rec#order_20">Index entries for linear recurrences with constant coefficients</a>, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 3148, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1).
%F A042581 G.f.: -(x^2-x-1)*(x^4-3*x^3+4*x^2-2*x+1)*(x^4-2*x^3+4*x^2-3*x+1)*(x^4+2*x^3+4*x^2+3*x+1)*(x^4+3*x^3+4*x^2+2*x+1) / (x^20-3148*x^10+1). - _Colin Barker_, Dec 18 2013
%t A042581 Denominator[Convergents[Sqrt[819], 30]] (* _Vincenzo Librandi_, Jan 25 2014 *)
%Y A042581 Cf. A042580, A040790.
%K A042581 nonn,frac,easy
%O A042581 0,3
%A A042581 _N. J. A. Sloane_.
%E A042581 More terms from _Colin Barker_, Dec 18 2013

%I A050762
%S A050762 1,1,2,3,5,8,13,21,34,89,610,987,1597,2584,4181,6765,10946,28657,
%T A050762 46368,75025,121393,196418,832040,1346269,2178309,3524578,14930352,
%U A050762 24157817,63245986,267914296,2971215073,4807526976,12586269025,86267571272
%N A050762 Fibonacci numbers (index numbers see A050761) containing no pair of consecutive equal digits (probably finite).
%C A050762 There are only 46 such numbers in the first 100,000 Fibonacci numbers. - _Harvey P. Dale_, Apr 01 2013
%H A050762 Harvey P. Dale, <a href="/A050762/b050762.txt">Table of n, a(n) for n = 1..46</a>
%t A050762 Select[Fibonacci[Range[100]],!MemberQ[Differences[IntegerDigits[#]],0]&] (* _Harvey P. Dale_, Apr 01 2013 *)
%Y A050762 Cf. A043096, A000045, A050761.
%Y A050762 See A000045 for the Fibonacci numbers.
%K A050762 nonn,base
%O A050762 1,3
%A A050762 _Patrick De Geest_, Sep 15 1999.

%I A065124
%S A065124 0,1,1,2,3,5,8,13,21,25,28,35,45,53,62,70,78,85,100,113,114,119,125,
%T A065124 136,144,154,163,173,183,194,206,220,228,232,244,251,261,269,278,295,
%U A065124 312,328,334,347,357,371,386,397,414,433,442,452,462,473,485,499,516
%N A065124 a(n) = (sum of digits of a(n-2)) + a(n-1); a(0) = 0 and a(1) = 1.
%H A065124 Harry J. Smith, <a href="/A065124/b065124.txt">Table of n, a(n) for n = 0..1000</a>
%p A065124 a[0]:=0:a[1]:=1:a[2]:=1:for n from 3 to 100 do d:=a[n-1]:s:=0:while d>0 do c:=d mod 10:s:=s+c:d:=(d-c)/10 od:e:=a[n-2]:t:=0:while e>0 do c:=e mod 10:t:=t+c:e:=(e-c)/10 od:a[n]:=a[n-1]+t od: seq(a[n], n=0..56); # _Zerinvary Lajos_, Mar 19 2009
%t A065124 a[0] = 0; a[1] = 1; a[n_] := a[n] = Apply[ Plus, IntegerDigits[ a[n - 2] ]] + a[n - 1]; Table[ a[n], {n, 0, 100} ]
%o A065124 (PARI) SumD(x)= { local(s=0); while (x>9, s+=x-10*(x\10); x\=10); return(s + x) } { for (n=0, 1000, if (n>1, a=SumD(a2) + a1; a2=a1; a1=a, if (n, a=a1=1, a=a2=0)); write("b065124.txt", n, " ", a) ) } \\ _Harry J. Smith_, Oct 10 2009
%Y A065124 Cf. A007612, A007953, A065076.
%K A065124 base,easy,nonn
%O A065124 0,4
%A A065124 _Robert G. Wilson v_, Nov 13 2001

%I A077373
%S A077373 0,1,1,2,3,5,8,13,21,34,55,89
%N A077373 Fibonacci numbers whose external digits as well as internal digits form a Fibonacci number.
%C A077373 Conjecture: sequence is finite. Is there any term > 89 in this sequence?
%H A077373 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/NonRecursions.html">Non Recursions</a>
%Y A077373 Cf. A077371, A077372.
%K A077373 base,more,nonn
%O A077373 1,4
%A A077373 _Amarnath Murthy_, Nov 06 2002

# Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE

# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/

Search: seq:1,1,2,3,5,8,13,21
Showing 41-50 of 92

%I A093091
%S A093091 1,1,2,3,5,8,13,21,34,55,89,14,4,10,3,18,14,13,21,32,27,34,53,59,61,
%T A093091 87,11,2,12,0,14,8,98,13,14,12,14,22,10,6,11,1,27,26,26,36,32,16,17,
%U A093091 12,28,53,52,62,68,48,33,29,40,81,10,5,11,4,13,0,11,6,81,62,69,12,1,91,15,16
%N A093091 "Fibonacci in pairs from left": start with a(1)=1, a(2)=1; repeatedly adjoin sum of previous two terms but chopped from the left into pairs of 2 digits.
%C A093091 Do all pairs of digits appear infinitely often? The sequence is not periodic.
%e A093091 ... a(11)=a(9)+a(10), a(12)=left pair of (a(10)+a(11)=55+89=14 4),
%e A093091 a(13)=right pair of (a(10)+a(11)=55+89=14 4),
%e A093091 a(14)=left pair of (a(11)+a(12)=89+14=10 3),
%e A093091 a(15)=right pair of (a(11)+a(12)=89+14=10 3), a(16)=a(12)+a(13) ...
%Y A093091 Cf. A093086, A093087, A093088, A093089, A093090.
%K A093091 nonn,base
%O A093091 1,3
%A A093091 _Bodo Zinser_, Mar 20 2004

%I A093332
%S A093332 0,1,1,2,3,5,8,13,21,34,56,92,152,251,414,684,1130,1868,3087,5102,
%T A093332 8433,13938,23038,38079,62940,104033,171955,284223,469789,776510,
%U A093332 1283487,2121464,3506550,5795947,9580072,15834821,26173243,43261534,71506628
%N A093332 a(0) = 0, a(1) = 1 and for n >=0, a(n+2) = int(sqrt(2 * (a(n)^2 + a(n+1)^2)))
%C A093332 First 10 terms are the same as the first 10 terms of the Fibonacci sequence (A000045).
%e A093332 a(5) = 5 because a(5) = int(sqrt(2* a(3)^2 + a(4)^2)) = int(sqrt(2*(2^2+3^2))) = int(sqrt(26) = 5.
%Y A093332 Cf. A000045, A093333, A093335.
%K A093332 easy,nonn
%O A093332 0,4
%A A093332 Robert A. Stump (rstump_2004(AT)yahoo.com), Apr 25 2004

%I A094102
%S A094102 1,1,1,1,1,1,2,1,1,1,1,2,3,2,1,1,1,1,2,3,5,3,2,1,1,1,1,2,3,5,8,5,3,2,
%T A094102 1,1,1,1,2,3,5,8,13,8,5,3,2,1,1,1,1,2,3,5,8,13,21,13,8,5,3,2,1,1,1,1,
%U A094102 2,3,5,8,13,21,34,21,13,8,5,3,2,1,1,1,1,2,3,5,8,13,21,34,55,34,21,13,8,5,3,2,1,1
%N A094102 Triangle read by rows in which row n contains Fib(1), ..., Fib(n-1), Fib(n), Fib(n-1), ..., Fib(1).
%e A094102 Triangle begins:
%e A094102 1
%e A094102 1 1 1
%e A094102 1 1 2 1 1
%e A094102 1 1 2 3 2 1 1
%e A094102 1 1 2 3 5 3 2 1 1
%e A094102 1 1 2 3 5 8 5 3 2 1 1
%Y A094102 Row sums are in A001911. Cf. A094103.
%K A094102 nonn,tabf
%O A094102 1,7
%A A094102 Alysia Veenhof (ladyluck1899(AT)hotmail.com), May 05 2004

%I A107358
%S A107358 0,1,1,2,3,5,8,13,21,34,55,89,144,233,376,608,982,1587,2564,4143,6694,
%T A107358 10816,17476,28237,45624,73717,119108,192449,310949,502416,811778,
%U A107358 1311630,2119265,3424201,5532650,8939375,14443788,23337539,37707610,60926041,98441202,159056294
%N A107358 Dying rabbits: a(n) = Fibonacci(n) for n <= 12; for n >= 13, a(n) = a(n-1) + a(n-2) - a(n-13).
%C A107358 In the limit, the growth rate is 1.61575... per generation as opposed to 1.61803... for Fibonacci numbers. - _T. D. Noe_, Jan 22 2009
%C A107358 If the rabbits die after 12 months, then those who were there in month 1 should die in month 13, whence a(13) = 144 + 89 - 1 = 232 and not 233. In month 14, no rabbits die because the only pair which was there in month 2 already dies. Then in month 15, the one pair born in month 3 will die. In general, the number of rabbits which die in month n (because they are aged 12 months) is equal to the number of newborn rabbits in month n - 12, which is the number of rabbits present in month n - 14. (Recall that a(n - 12) = a(n - 13) + a(n - 14) - #(dying rabbits) = #(rabbits from previous month) + #(newborn rabbits) - #(dying rabbits).) So the recurrence should read a(n) = a(n - 1) + a(n - 2) - a(n - 14). - _M. F. Hasler_, Oct 06 2017
%H A107358 T. D. Noe, <a href="/A107358/b107358.txt">Table of n, a(n) for n = 0..500</a>
%H A107358 J. H. E. Cohn, <a href="http://www.fq.math.ca/Scanned/2-2/cohn1.pdf">Letter to the editor</a>, Fib. Quart. 2 (1964), 108.
%H A107358 V. E. Hoggatt, Jr. and D. A. Lind, <a href="http://www.fq.math.ca/Scanned/7-5/hoggatt.pdf">The dying rabbit problem</a>, Fib. Quart. 7 (1969), 482-487.
%H A107358 <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,0,0,0,0,0,0,0,0,0,-1).
%F A107358 G.f.: x/((x-1)*(1+x)*(x^11+x^9+x^7+x^5+x^3+x-1)). - _R. J. Mathar_, Jul 27 2009
%p A107358 with(combinat); f:=proc(n) option remember; if n <= 12 then RETURN(fibonacci(n)); fi; f(n-1)+f(n-2)-f(n-13); end;
%t A107358 LinearRecurrence[{1,1,0,0,0,0,0,0,0,0,0,0,-1},Fibonacci[Range[0,12]],50] (* _Harvey P. Dale_, Feb 28 2013 *)
%o A107358 (PARI) Vec(x/(x^13-x^2-x+1)+O(x^99)) \\ _Charles R Greathouse IV_, Jun 10 2011
%Y A107358 See A000045 for the Fibonacci numbers. This is a better version of A000044.
%K A107358 nonn,easy
%O A107358 0,4
%A A107358 _N. J. A. Sloane_, May 25 2005

%I A121104
%S A121104 1,1,2,3,5,8,13,21,5,24,15,21,18,1,27,43,11,38,63,49,44,32,88,2,83,59,
%T A121104 73,76,79,63,113,9,94,61,6,123,76,149,127,34,74,124,32,83,1,3,91,212,
%U A121104 204,232,85,76,171,141,148,234,145,38,2,271,208,179,194,73,14,127,177
%N A121104 a(n) = Fibonacci(n - 1) modulo the n-th prime number.
%H A121104 Vincenzo Librandi, <a href="/A121104/b121104.txt">Table of n, a(n) for n = 2..5000</a>
%F A121104 a(n) = Fibonacci(n - 1) modulo Prime(n).
%e A121104 a(10)=5 because the 9th Fibonacci=34, the 10th Prime=29, and 34 mod 29=5.
%t A121104 With[{nn=70},Mod[First[#],Last[#]]&/@Thread[{Fibonacci[Range[nn-1]], Prime[ Range[2,nn]]}]] (* _Harvey P. Dale_, Feb 27 2013 *)
%t A121104 Table[Mod[Fibonacci[n - 1], Prime[n]], {n, 2, 70}] (* _Vincenzo Librandi_, Jun 19 2017 *)
%o A121104 (PARI) a(n) = fibonacci(n-1) % prime(n); \\ _Michel Marcus_, Jun 18 2017
%o A121104 (PARI) fibmod(n, m)=((Mod([1, 1; 1, 0], m))^n)[1, 2]
%o A121104 a(n)=lift(fibmod(n-1,prime(n))) \\ _Charles R Greathouse IV_, Jun 19 2017
%o A121104 (MAGMA) [Fibonacci(n-1) mod NthPrime(n): n in [2..70]]; // _Vincenzo Librandi_, Jun 19 2017
%Y A121104 Cf. A000040, A000045, A072123.
%K A121104 nonn
%O A121104 2,3
%A A121104 _Gil Broussard_, Aug 12 2006

%I A147316
%S A147316 -6765,4181,-2584,1597,-987,610,-377,233,-144,89,-55,34,-21,13,-8,5,
%T A147316 -3,2,-1,1,0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,
%U A147316 4181,6765,10946,17711,28657,46368,75025,121393,196418,317811,514229,832040
%N A147316 Fibonacci numbers (A000045) starting at offset -20.
%C A147316 The recurrence relation a(n+1) = a(n) + a(n-1) defines the Fibonacci sequence for all (positive and negative) integer indices, given any two values with indices of opposite parity, e.g., a(0) and a(1), or a(-1) and a(42). Any other Fibonacci-type sequence {b(n)} satisfying this recurrence relation can be written as b(n) = b(1)*A000045(n) + b(0)*A000045(n-1). This can be seen from the fact that the set of all sequences satisfying a given linear recurrence relation of order 2 with constant coefficients forms a vector space of dimension two. So each element (sequence) of this space is a linear combination of any two elements which are not proportional to each other and thus form a base. The most natural choice of such a base could be the two sequences having (b(0), b(1)) = (0, 1) resp (1, 0). These are A000045 and n -> A000045(n-1) = A212804 (extended to negative indices, if needed). - _M. F. Hasler_, May 10 2017
%H A147316 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1).
%H A147316 Philipp Fahr and Claus Michael Ringel, <a href="http://www.mathematik.uni-bielefeld.de/~ringel/opus/fr-zwei.pdf">Categorification of the Fibonacci Numbers Using Representations of Quivers</a>
%F A147316 a(n) = a(n-1) + a(n-2). - _R. J. Mathar_, Nov 30 2008
%t A147316 Array[Fibonacci, 51, -20] (* _Michael De Vlieger_, May 10 2017 *)
%o A147316 (PARI) a(n)=fibonacci(n) \\ _M. F. Hasler_, May 10 2017
%Y A147316 Cf. A000285, A022113, A001060.
%Y A147316 Of course A000045 is the main entry for the Fibonacci numbers.
%Y A147316 See also A039834 for A000045 extended to negative indices.
%K A147316 sign,easy,less
%O A147316 -20,1
%A A147316 _Roger L. Bagula_, Nov 05 2008
%E A147316 Extended to n = -20 .. 30 by _M. F. Hasler_, May 10 2017

%I A177372
%S A177372 1,1,2,3,5,8,13,21,34,55,89,144,233,610,2584,4181,10946,46368,121393,
%T A177372 196418,514229,832040,1346269,14930352,39088169,63245986,102334155,
%U A177372 165580141,1836311903,12586269025,32951280099,139583862445,365435296162
%N A177372 Fibonacci numbers whose decimal expansion does not contain any digit 7
%C A177372 Probability that Fib(n) contains no digit 7 goes to zero as n grows to infinity. I suppose that the maximum number is Fib(224) having 47 digits.
%e A177372 a(14)=610 since it is the 14th Fibonacci containing no 7's.
%t A177372 Select[Fibonacci[Range[100]],DigitCount[#,10,7]==0&] (* _Harvey P. Dale_, Dec 13 2014 *)
%Y A177372 Cf. A000045, A177194, A177195, A177231, A177245, A177246, A176253, A177247
%K A177372 nonn,base
%O A177372 1,3
%A A177372 _Carmine Suriano_, May 07 2010

%I A206139
%S A206139 1,1,1,2,3,5,8,13,21,34,55,88,141,224,356,563,890,1401,2202,3448,5386,
%T A206139 8386,13025,20175,31180,48077,73976,113588,174057,266174,406224,
%U A206139 618729,940552,1427038,2161122,3266956,4930052,7427314,11171332,16776169,25154204
%N A206139 G.f.: A(x) = Sum_{n>=0} x^(n*(n+1)/2) / Product_{k=1..n} (1-x^k)^(n-k+1).
%H A206139 Vaclav Kotesovec, <a href="/A206139/b206139.txt">Table of n, a(n) for n = 0..400</a>
%e A206139 G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 8*x^6 + 13*x^7 +...
%e A206139 where
%e A206139 A(x) = 1 + x/(1-x) + x^3/((1-x)^2*(1-x^2)) + x^6/((1-x)^3*(1-x^2)^2*(1-x^3)) + x^10/((1-x)^4*(1-x^2)^3*(1-x^3)^2*(1-x^4)) +...
%o A206139 (PARI) {a(n)=polcoeff(sum(m=0,n,x^(m*(m+1)/2)/prod(k=1,m,(1-x^k +x*O(x^n))^(m-k+1))),n)}
%o A206139 for(n=0,61,print1(a(n),", "))
%Y A206139 Cf. A206119.
%K A206139 nonn
%O A206139 0,4
%A A206139 _Paul D. Hanna_, Feb 04 2012

%I A217737
%S A217737 1,1,2,3,5,8,13,21,34,55,89,144,51,167,130,171,67,190,1,45,320,1,505,
%T A217737 168,275,649,614,319,59,620,125,837,376,407,485,1296,1331,419,466,
%U A217737 1435,1231,1420,1289,1653,830,2069,2161,1344,1849,1975,746,1167,1589,872,2645,2205
%N A217737 a(n) = Fibonacci(n) mod n*(n+1).
%H A217737 Alois P. Heinz, <a href="/A217737/b217737.txt">Table of n, a(n) for n = 1..10000</a>
%F A217737 A000045(n) modulo A002378(n).
%p A217737 a:= proc(n) local r, M, p, m; r, M, p, m:=
%p A217737       <<1|0>, <0|1>>, <<0|1>, <1|1>>, n, n*(n+1);
%p A217737       do if irem(p, 2, 'p')=1 then r:= r.M mod m fi;
%p A217737          if p=0 then break fi; M:= M.M mod m
%p A217737       od; r[1, 2]
%p A217737     end:
%p A217737 seq(a(n), n=1..100);  # _Alois P. Heinz_, Nov 26 2016
%t A217737 Table[Mod[Fibonacci[n],n(n+1)],{n,60}] (* _Harvey P. Dale_, Oct 02 2017 *)
%o A217737 (Python)
%o A217737 prpr, prev = 0, 1
%o A217737 for i in range(1, 333):
%o A217737     cur = prpr + prev
%o A217737     print str(prev % (i*(i+1))) + ',',
%o A217737     prpr, prev = prev, cur
%o A217737 (PARI) a(n)=fibonacci(n)%(n*(n+1)) \\ _Charles R Greathouse IV_, Jun 23 2017
%Y A217737 Cf. A000045, A002708, A121343, A132634, A132636.
%K A217737 nonn,easy
%O A217737 1,3
%A A217737 _Alex Ratushnyak_, Mar 22 2013

%I A273715
%S A273715 0,1,1,1,2,3,5,8,13,21,1,34,57,6,90,158,27,241,445,107,1,652,1269,396,
%T A273715 10,1780,3655,1404,66,4899,10611,4838,356,1,13581,31002,16344,1700,15,
%U A273715 37893,91048,54429,7482,135,106340,268536,179332,31070,940,1
%N A273715 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k peaks of width 1 (i.e., UHD configurations, where U=(0,1), H(1,0), D=(0,-1)), (n>=2, k>=0).
%C A273715 Sum of entries in row n = A082582(n).
%C A273715 Sum(k*T(n,k),k>=1) = A273716(n).
%H A273715 Alois P. Heinz, <a href="/A273715/b273715.txt">Rows n = 2..250, flattened</a>
%H A273715 M. Bousquet-Mélou and A. Rechnitzer <a href="http://dx.doi.org/10.1016/S0196-8858(02)00553-5">The site-perimeter of bargraphs</a> Adv. Appl. Math., 31, 2003, 86-112.
%H A273715 Emeric Deutsch, S Elizalde, <a href="http://arxiv.org/abs/1609.00088">Statistics on bargraphs viewed as cornerless Motzkin paths</a>, arXiv preprint arXiv:1609.00088, 2016
%F A273715 G.f.: G(t,z) satisfies z*G^2 - (1-2*z-z^2-z^3+t*z^3)G + z^2*(t+z-t*z) = 0.
%e A273715 Row 4 is 2,3 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3] which, clearly, have 0,1,1,0,1 peaks of width 1.
%e A273715 Triangle T(n,k) begins:
%e A273715 :    0,    1;
%e A273715 :    1,    1;
%e A273715 :    2,    3;
%e A273715 :    5,    8;
%e A273715 :   13,   21,   1;
%e A273715 :   34,   57,   6;
%e A273715 :   90,  158,  27;
%e A273715 :  241,  445, 107,  1;
%e A273715 :  652, 1269, 396, 10;
%p A273715 eq := z*G^2-(1-2*z-z^2-z^3+t*z^3)*G+z^2*(t+z-t*z) = 0: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 25)): for n from 2 to 20 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 2 to 20 do seq(coeff(P[n], t, j), j = 0 .. degree(P[n])) end do; # yields sequence in triangular form
%p A273715 # second Maple program:
%p A273715 b:= proc(n, y, t, h) option remember; expand(
%p A273715       `if`(n=0, (1-t)*z^h, `if`(t<0, 0, b(n-1, y+1, 1, 0))+
%p A273715       `if`(t>0 or y<2, 0, b(n, y-1, -1, 0)*z^h)+
%p A273715       `if`(y<1, 0, b(n-1, y, 0, `if`(t>0, 1, 0)))))
%p A273715     end:
%p A273715 T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$3)):
%p A273715 seq(T(n), n=2..20);  # _Alois P. Heinz_, Jun 06 2016
%t A273715 b[n_, y_, t_, h_] := b[n, y, t, h] = Expand[ If[n == 0, (1 - t)*z^h, If[t < 0, 0, b[n - 1, y + 1, 1, 0]] + If[t > 0 || y < 2, 0, b[n, y - 1, -1, 0]*z^h] + If[y < 1, 0, b[n - 1, y, 0, If[t > 0, 1, 0]]]]] ; T[n_] := Function [p, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[n, 0, 0, 0]]; Table[T[n], {n, 2, 20}] // Flatten (* _Jean-François Alcover_, Nov 29 2016 after _Alois P. Heinz_ *)
%Y A273715 Cf. A082582, A273716.
%K A273715 nonn,tabf
%O A273715 2,5
%A A273715 _Emeric Deutsch_, May 28 2016

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# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/

Search: seq:1,1,2,3,5,8,13,21
Showing 51-60 of 92

%I A302019
%S A302019 1,1,2,3,5,8,13,21,34,56,91,149,243,397,648,1058,1727,2819,4602,7512,
%T A302019 12263,20018,32678,53344,87080,142151,232050,378803,618366,1009433,
%U A302019 1647819,2689933,4391101,7168122,11701387,19101580,31181804,50901806,83093134,135642908,221426218,361460624
%N A302019 Expansion of 1/(1 - x*Sum_{k>=0} x^(k^3)).
%H A302019 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A302019 <a href="/index/Su#ssq">Index entries for sequences related to sums of cubes</a>
%F A302019 G.f.: 1/(1 - x*Sum_{k>=0} x^(k^3)).
%F A302019 a(0) = 1; a(n) = Sum_{k=1..n} A010057(k-1)*a(n-k).
%t A302019 nmax = 41; CoefficientList[Series[1/(1 - x Sum[x^k^3, {k, 0, nmax}]), {x, 0, nmax}], x]
%Y A302019 Antidiagonal sums of A290054.
%Y A302019 Cf. A010057, A302018.
%K A302019 nonn
%O A302019 0,3
%A A302019 _Ilya Gutkovskiy_, Mar 30 2018

%I A304790
%S A304790 1,1,2,3,5,8,13,21,36,55,95,149,281,430,781,1211,2245,3456,6728,10092,
%T A304790 18061,31529,51378,85659,167089,252748,431819,817991,1292697
%N A304790 The maximal number of different domino tilings allowed by the Ferrers-Young diagram of a single partition of 2n.
%H A304790 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FerrersDiagram.html">Ferrers Diagram</a>
%H A304790 Wikipedia, <a href="https://en.wikipedia.org/wiki/Domino_(mathematics)">Domino</a>
%H A304790 Wikipedia, <a href="https://en.wikipedia.org/wiki/Domino_tiling">Domino tiling</a>
%H A304790 Wikipedia, <a href="https://en.wikipedia.org/wiki/Ferrers_diagram">Ferrers diagram</a>
%H A304790 Wikipedia, <a href="https://en.wikipedia.org/wiki/Mutilated_chessboard_problem">Mutilated chessboard problem</a>
%H A304790 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%H A304790 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau#Diagrams">Young tableau, Diagrams</a>
%F A304790 a(n) = max { k : A304789(n,k) > 0 }.
%F A304790 a(A001105(n)) = A004003(n).
%F A304790 a(n) = A000045(n+1) for n < 8.
%e A304790 a(11) = 149 different domino tilings are possible for 444442 and 6655.
%e A304790 a(18) = 6728 different domino tilings are possible for 666666.
%p A304790 h:= proc(l, f) option remember; local k; if min(l[])>0 then
%p A304790      `if`(nops(f)=0, 1, h(map(u-> u-1, l[1..f[1]]), subsop(1=[][], f)))
%p A304790     else for k from nops(l) while l[k]>0 by -1 do od;
%p A304790         `if`(nops(f)>0 and f[1]>=k, h(subsop(k=2, l), f), 0)+
%p A304790         `if`(k>1 and l[k-1]=0, h(subsop(k=1, k-1=1, l), f), 0)
%p A304790       fi
%p A304790     end:
%p A304790 g:= l-> `if`(add(`if`(l[i]::odd, (-1)^i, 0), i=1..nops(l))=0,
%p A304790         `if`(l=[], 1, h([0$l[1]], subsop(1=[][], l))), 0):
%p A304790 b:= (n, i, l)-> `if`(n=0 or i=1, g([l[], 1$n]), max(b(n, i-1, l),
%p A304790                    b(n-i, min(n-i, i), [l[], i]))):
%p A304790 a:= n-> b(2*n$2, []):
%p A304790 seq(a(n), n=0..15);
%Y A304790 Cf. A000045, A001105, A004003, A304789.
%K A304790 nonn
%O A304790 0,3
%A A304790 _Alois P. Heinz_, May 18 2018

%I A306486
%S A306486 0,1,1,2,3,5,8,13,21,36,58,96,159,262,431,712,1172,1934,3189,5256,
%T A306486 8667,14289,23559,38841,64039,105583,174076,287003,473188,780155,
%U A306486 1286258,2120681,3496412,5764609,9504233,15669832,25835185,42595018,70227313,115785266
%N A306486 Number of squares in the interval [e^(n-1), e^n).
%C A306486 The lower endpoint e^(n-1) is included; the upper endpoint is not included. The terms a(0) to a(8) coincide with the Fibonacci numbers.
%H A306486 Alois P. Heinz, <a href="/A306486/b306486.txt">Table of n, a(n) for n = 0..4607</a> (first 501 terms from Alexei Kourbatov)
%F A306486 a(n) = ceiling(sqrt(exp(n))) - ceiling(sqrt(exp(n-1))).
%F A306486 From _Alois P. Heinz_, Feb 19 2019: (Start)
%F A306486 Lim_{n->oo} a(n+1)/a(n) = sqrt(e) = 1.64872127... = A019774.
%F A306486 a(n) = A005181(n+1) - A005181(n). (End)
%F A306486 a(n) = (1-1/sqrt(e))*e^(n/2)+O(1) ~ 0.39346934...*e^(n/2) ~ A290506*e^(n/2). - _Alexei Kourbatov_, Feb 20 2019
%e A306486 Between exp(2) and exp(3) there are two squares, namely, 9 and 16; therefore, a(3)=2.
%p A306486 a:= n-> (f-> f(n)-f(n-1))(i-> ceil(exp(i/2))):
%p A306486 seq(a(n), n=0..44);  # _Alois P. Heinz_, Feb 18 2019
%o A306486 (PARI) a(n)=ceil(sqrt(exp(n)))-ceil(sqrt(exp(n-1)));
%o A306486 for(n=0,50,print1(a(n)", "))
%Y A306486 Cf. A000045, A000290, A001113, A005181, A019774, A290506, A306604.
%K A306486 nonn
%O A306486 0,4
%A A306486 _Alexei Kourbatov_, Feb 18 2019

%I A324969
%S A324969 1,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,
%T A324969 10946,17711,28657,46368
%N A324969 Number of unlabeled rooted identity trees with n vertices whose non-leaf terminal subtrees are all different.
%C A324969 A rooted identity tree is an unlabeled rooted tree with no repeated branches directly under the same root. This sequence counts rooted identity trees satisfying the additional condition that all non-leaf terminal subtrees are different.
%C A324969 Appears to be essentially the same as the Fibonacci sequence A000045. - _R. J. Mathar_, Mar 28 2019
%e A324969 The a(1) = 1 through a(7) = 8 trees:
%e A324969   o  (o)  ((o))  (o(o))   ((o(o)))   (o(o(o)))    ((o(o(o))))
%e A324969                  (((o)))  (o((o)))   (((o(o))))   (o((o(o))))
%e A324969                           ((((o))))  ((o((o))))   (o(o((o))))
%e A324969                                      (o(((o))))   ((((o(o)))))
%e A324969                                      (((((o)))))  (((o((o)))))
%e A324969                                                   ((o(((o)))))
%e A324969                                                   (o((((o)))))
%e A324969                                                   ((((((o))))))
%t A324969 durtid[n_]:=Join@@Table[Select[Union[Sort/@Tuples[durtid/@ptn]],UnsameQ@@#&&UnsameQ@@Cases[#,{__},{0,Infinity}]&],{ptn,IntegerPartitions[n-1]}];
%t A324969 Table[Length[durtid[n]],{n,15}]
%Y A324969 The Matula-Goebel numbers of these trees are given by A324968.
%Y A324969 Cf. A000081, A004111, A276625, A290689, A317713, A324923, A324936, A324971, A324978.
%K A324969 nonn,more
%O A324969 1,4
%A A324969 _Gus Wiseman_, Mar 21 2019

%I A005170 M0694
%S A005170 1,0,1,1,2,3,5,8,13,21,35,55,93,149,248,403,670,1082
%N A005170 Erroneous version of A226999.
%D A005170 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005170 R. K. Guy, <a href="/A005169/a005169_6.pdf">Letter to N. J. A. Sloane</a>, Sep 25 1986.
%H A005170 R. K. Guy, <a href="/A005728/a005728.pdf">Letter to N. J. A. Sloane, 1987</a>
%H A005170 R. K. Guy, <a href="http://www.jstor.org/stable/2322249">The strong law of small numbers</a>. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
%H A005170 R. K. Guy, <a href="/A005165/a005165.pdf">The strong law of small numbers</a>. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
%K A005170 dead
%O A005170 1,5

%I A013986
%S A013986 1,0,1,1,2,3,5,8,13,21,33,54,86,139,223,359,577,928,1492,2399,3858,
%T A013986 6203,9975,16039,25791,41471,66685,107228,172421,277250,445813,716860,
%U A013986 1152698,1853519,2980426,4792474,7706215
%N A013986 Expansion of 1/(1-x^2-x^3-x^4-x^5-x^6-x^7-x^8-x^9).
%C A013986 Number of compositions of n into parts p where 2 <= p < = 9. [_Joerg Arndt_, Jun 24 2013]
%H A013986 Vincenzo Librandi, <a href="/A013986/b013986.txt">Table of n, a(n) for n = 0..1000</a>
%H A013986 R. Mullen, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Mullen/mullen2.html">On Determining Paint by Numbers Puzzles with Nonunique Solutions</a>, JIS 12 (2009) 09.6.5
%H A013986 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0, 1, 1, 1, 1, 1, 1, 1, 1).
%F A013986 a(0)=1, a(1)=0, a(2)=1, a(3)=1, a(4)=2, a(5)=3, a(6)=5, a(7)=8, a(8)=13, a(n)=a(n-2)+a(n-3)+a(n-4)+a(n-5)+a(n-6)+a(n-7)+a(n-8)+a(n-9). - _Harvey P. Dale_, Dec 17 2013
%t A013986 CoefficientList[Series[1 / (1 - x^2 - x^3 - x^4 - x^5 - x^6 - x^7 - x^8 - x^9), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jun 24 2013 *)
%t A013986 CoefficientList[Series[1/(1-Total[x^Range[2,9]]),{x,0,40}],x] (* or *) LinearRecurrence[{0,1,1,1,1,1,1,1,1},{1,0,1,1,2,3,5,8,13},40] (* _Harvey P. Dale_, Dec 17 2013 *)
%o A013986 (MAGMA) m:=40; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-x^2-x^3-x^4-x^5-x^6-x^7-x^8-x^9))); // _Vincenzo Librandi_, Jun 24 2013
%Y A013986 See A000045 for the Fibonacci numbers.
%K A013986 nonn,easy
%O A013986 0,5
%A A013986 _N. J. A. Sloane_.

%I A013987
%S A013987 1,0,1,1,2,3,5,8,13,21,34,54,88,141,228,367,592,954,1538,2479,3996,
%T A013987 6441,10383,16736,26978,43486,70097,112991,182134,293587,473242,
%U A013987 762833,1229634,1982084,3194982,5150088,8301584,13381575,21570168,34769609,56046190
%N A013987 Expansion of 1/(1-x^2-x^3-x^4-x^5-x^6-x^7-x^8-x^9-x^10).
%C A013987 Number of compositions of n into parts p where 2 <= p < = 10. [_Joerg Arndt_, Jun 24 2013]
%H A013987 Vincenzo Librandi, <a href="/A013987/b013987.txt">Table of n, a(n) for n = 0..1000</a>
%H A013987 R. Mullen, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Mullen/mullen2.html">On Determining Paint by Numbers Puzzles with Nonunique Solutions</a>, JIS 12 (2009) 09.6.5
%t A013987 CoefficientList[Series[1 / (1 - x^2 - x^3 - x^4 - x^5 - x^6 - x^7 - x^8 - x^9 - x^10), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jun 24 2013 *)
%o A013987 (MAGMA) m:=40; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-x^2-x^3-x^4-x^5-x^6-x^7-x^8-x^9-x^10))); // _Vincenzo Librandi_, Jun 24 2013
%Y A013987 See A000045 for the Fibonacci numbers.
%K A013987 nonn,easy
%O A013987 0,5
%A A013987 _N. J. A. Sloane_.

%I A023441
%S A023441 0,1,1,2,3,5,8,13,21,34,55,89,143,231,372,600,967,1559,2513,4051,6530,
%T A023441 10526,16967,27350,44086,71064,114550,184647,297638,479772,773359,
%U A023441 1246601,2009434,3239068,5221152,8416134
%N A023441 Dying rabbits: a(n) = a(n-1) + a(n-2) - a(n-11).
%H A023441 J. H. E. Cohn, <a href="http://www.fq.math.ca/Scanned/2-2/cohn1.pdf">Letter to the editor</a>, Fib. Quart. 2 (1964), 108.
%H A023441 V. E. Hoggatt, Jr. and D. A. Lind, <a href="http://www.fq.math.ca/Scanned/7-5/hoggatt.pdf">The dying rabbit problem</a>, Fib. Quart. 7 (1969), 482-487.
%F A023441 G.f.: x/((x-1)*(1+x)*(x^9+x^7+x^5+x^3+x-1)). [_R. J. Mathar_, Jul 27 2009]
%o A023441 (PARI) concat(0, Vec(x/((x-1)*(1+x)*(x^9+x^7+x^5+x^3+x-1)) + O(x^50))) \\ _Michel Marcus_, Sep 06 2017
%Y A023441 See A000045 for the Fibonacci numbers.
%K A023441 nonn
%O A023441 0,4
%A A023441 _N. J. A. Sloane_.

%I A023442
%S A023442 0,1,1,2,3,5,8,13,21,34,55,89,144,232,375,605,977,1577,2546,4110,6635,
%T A023442 10711,17291,27913,45060,72741,117426,189562,306011,493996,797461,
%U A023442 1287347,2078173,3354809,5415691,8742587
%N A023442 Dying rabbits: a(n) = a(n-1) + a(n-2) - a(n-12).
%H A023442 J. H. E. Cohn, <a href="http://www.fq.math.ca/Scanned/2-2/cohn1.pdf">Letter to the editor</a>, Fib. Quart. 2 (1964), 108.
%H A023442 V. E. Hoggatt, Jr. and D. A. Lind, <a href="http://www.fq.math.ca/Scanned/7-5/hoggatt.pdf">The dying rabbit problem</a>, Fib. Quart. 7 (1969), 482-487.
%F A023442 G.f.: x/ ( (x-1)*(x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2-1) ). - _R. J. Mathar_, Nov 29 2011
%o A023442 (PARI) concat(0, Vec(x/ ( (x-1)*(x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2-1) ) + O(x^50))) \\ _Michel Marcus_, Sep 06 2017
%Y A023442 See A000045 for the Fibonacci numbers.
%K A023442 nonn
%O A023442 0,4
%A A023442 _N. J. A. Sloane_.

%I A053412
%S A053412 1,1,2,3,5,8,13,21,17711,317811
%N A053412 n-th nonzero Fibonacci numbers arising in A053408.
%Y A053412 See A000045 for the Fibonacci numbers.
%K A053412 nonn
%O A053412 0,3
%A A053412 _G. L. Honaker, Jr._, Jan 09 2000

# Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE

# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/

Search: seq:1,1,2,3,5,8,13,21
Showing 61-70 of 92

%I A055806
%S A055806 1,1,1,2,3,5,8,13,21,33,53,79,125,176,273,365,554,709,1053,1300,1891,
%T A055806 2267,3234,3785,5303,6085,8385,9465,12845,14302,19139,21065,27828,
%U A055806 30329,39593,42790,55251,59281,75772
%N A055806 T(n,n-6), array T as in A055801.
%K A055806 nonn
%O A055806 6,4
%A A055806 _Clark Kimberling_, May 28 2000

%I A117502
%S A117502 1,1,2,1,1,3,1,1,2,5,1,1,2,3,8,1,1,2,3,5,13,1,1,2,3,5,8,21,1,1,2,3,5,
%T A117502 8,13,34,1,1,2,3,5,8,13,21,55,1,1,2,3,5,8,13,21,34,89
%N A117502 Triangle, row sums = A001595.
%C A117502 Row sums = A001595 starting (1, 3, 9, 15, 25, 41...)
%F A117502 n-th row = first n Fibonacci terms, with a deletion of F(n). Columns of the triangle are difference terms of the array in A117501.
%e A117502 Row 5 of the triangle = (1, 1, 2, 3, 8); the first 5 Fibonacci terms with a deletion of F(5) = 5.
%e A117502 First few rows of the triangle are:
%e A117502 1;
%e A117502 1, 2;
%e A117502 1, 1, 3;
%e A117502 1, 1, 2, 5;
%e A117502 1, 1, 2, 3, 8;
%e A117502 ...
%Y A117502 Cf. A117501, A001595.
%K A117502 nonn,tabl
%O A117502 1,3
%A A117502 _Gary W. Adamson_, Mar 23 2006

%I A134404
%S A134404 0,0,1,0,0,1,1,1,0,0,1,1,2,1,1,0,0,1,1,2,3,2,1,1,0,0,1,1,2,3,5,3,2,1,
%T A134404 1,0,0,1,1,2,3,5,8,5,3,2,1,1,0,0,1,1,2,3,5,8,13,8,5,3,2,1,1,0,0,1,1,2,
%U A134404 3,5,8,13,21,13,8,5,3,2,1,1,0,0,1,1,2,3,5,8,13,21,34,21,13,8,5
%N A134404 Triangle read by rows in which row n contains Fib(0), ..., Fib(n-1), Fib(n), Fib(n-1), ..., Fib(0).
%e A134404 Triangle begins:
%e A134404 0
%e A134404 0 1 0
%e A134404 0 1 1 1 0
%e A134404 0 1 1 2 1 1 0
%e A134404 0 1 1 2 3 2 1 1 0
%e A134404 0 1 1 2 3 5 3 2 1 1 0
%e A134404 0 1 1 2 3 5 8 5 3 2 1 1 0
%e A134404 0 1 1 2 3 5 8 13 8 5 3 2 1 1 0
%e A134404 0 1 1 2 3 5 8 13 21 13 8 5 3 2 1 1 0
%e A134404 0 1 1 2 3 5 8 13 21 34 21 13 8 5 3 2 1 1 0
%e A134404 0 1 1 2 3 5 8 13 21 34 55 34 21 13 8 5 3 2 1 1 0
%e A134404 0 1 1 2 3 5 8 13 21 34 55 89 55 34 21 13 8 5 3 2 1 1 0
%Y A134404 Cf. A094102.
%K A134404 nonn,tabf
%O A134404 0,13
%A A134404 _N. J. A. Sloane_, Apr 07 2008

%I A137290
%S A137290 1,1,2,3,5,8,13,21,4,25,29,24,23,17,10,27,7,4,11,15,26,11,7,18,25,13,
%T A137290 8,21,29,20,19,9,28,7,5,12,17,29,16,15,1,16,17,3,20,23,13,6,19,25,14,
%U A137290 9,23,2,25,27,22,19,11,0,11,11,22,3,25,28,23,21,14,5,19,24,13,7,20,27,17,14
%N A137290 Fibonacci(n) mod 30.
%C A137290 Has period 120.
%H A137290 Michel Marcus, <a href="/A137290/b137290.txt">Table of n, a(n) for n = 1..130</a>
%H A137290 <a href="/index/Rec#order_120">Index entries for linear recurrences with constant coefficients</a>, order 120.
%o A137290 (PARI) a(n) = fibonacci(n) % 30 \\ _Michel Marcus_, Jun 12 2013
%Y A137290 Cf. A000045, A003893, A007887, A079345, A105471.
%K A137290 nonn
%O A137290 1,3
%A A137290 Aaron M. Churchill (churchil(AT)math.udel.edu), Mar 15 2008

%I A147659
%S A147659 1,1,2,3,5,8,13,21,34,55,90,146,237,385,625,1015,1648,2676,4345,7055,
%T A147659 11456,18602,30205,49046,79639,129315,209977,340953,553627,898959,
%U A147659 1459697,2370203,3848649,6249296,10147379,16476944,26754661,43443243
%N A147659 Expansion of 1/(1-x-x^2-x^10+x^12).
%C A147659 Coefficient expansion of toral of inverse of low ratio (1.6237635378007012) Pisot Polynomial: a(n)=Coefficient_Expansion(1/( 1 - x^2 - x^10 - x^11 + x^12)). [Original definition]
%C A147659 The 1 + x^2 - x^10 - x^11 + x^12, is not Pisot, so x^22 doubling is the limit that sequence of polynomials below the Golden mean ratio.
%H A147659 Vincenzo Librandi, <a href="/A147659/b147659.txt">Table of n, a(n) for n = 0..1000</a>
%H A147659 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1).
%t A147659 f[x_] = 1 - x^2 - x^10 - x^11 + x^12; g[x] = ExpandAll[x^12*f[1/x]]; a = Table[SeriesCoefficient[Series[1/g[x], {x, 0, 50}], n], {n, 0, 50}]
%t A147659 CoefficientList[Series[1/(1-x-x^2-x^10+x^12),{x,0,40}],x] (* _Harvey P. Dale_, May 31 2012 *)
%K A147659 nonn
%O A147659 0,3
%A A147659 _Roger L. Bagula_, Nov 09 2008

%I A147660
%S A147660 1,1,2,3,5,8,13,21,34,54,87,140,225,362,582,936,1505,2420,3892,6259,
%T A147660 10065,16186,26029,41858,67313,108248,174077,279938,450176,723941,
%U A147660 1164190,1872167,3010685,4841568,7785863,12520667,20134840,32379408
%N A147660 Coefficient expansion of toral of inverse of low ratio (1.6081283851873882) Pisot Polynomial: a(n)=Coefficient_Expansion(1/( -1 + x^2 - x^9 - x^10 + x^11)).
%C A147660 The next 1 + x^2 - x^10 - x^11 + x^12, is not Pisot, so x^11 is the limit that sequence of polynomials below the Golden mean ratio.
%F A147660 a(n)=Coefficient_Expansion(1/( -1 + x^2 - x^9 - x^10 + x^11)).
%t A147660 f[x_] = -1 + x^2 - x^9 - x^10 + x^11; g[x] = ExpandAll[x^11*f[1/x]]; a = Table[SeriesCoefficient[Series[1/g[x], {x, 0, 50}], n], {n, 0, 50}]
%K A147660 nonn
%O A147660 0,3
%A A147660 _Roger L. Bagula_, Nov 09 2008

%I A191869
%S A191869 0,0,1,1,2,3,5,8,13,21,34,55,88,143,231,373,603,974,1574,2543,4109,
%T A191869 6639,10727,17332,28004,45248,73109,118126,190862,308385,498273,
%U A191869 805084,1300814,2101789,3395964,5487026,8865658,14324680,23145090,37396661,60423625
%N A191869 First differences of the dying rabbits sequence A000044.
%F A191869 G.f.: x^3(1 + x + x^2 + x^3 + x^4)(1 - x + x^2 - x^3 + x^4)/(1 - x - x^3 - x^5 - x^7 - x^9 - x^11). - _Charles R Greathouse IV_, Jun 19 2011
%t A191869 A000044 = CoefficientList[Series[1/(1 - z - z^3 - z^5 - z^7 - z^9 - z^11), {z, 0, 200}], z]; GetDiff[seq_List] := Drop[seq, 1] - Drop[seq, -1]; A191869 = GetDiff[A000044]
%o A191869 (PARI) A191869_list=Vec((-x^11-x^9-x^7-x^5-x^3)/(x^11+x^9+x^7+x^5+x^3+x-1)+O(x^99)) /* returns a list of the first 96 nonzero terms, a(3)...a(99) */
%o A191869 (PARI) A191869(n)=polcoeff((1+x^2+x^4+x^6+x^8)/(1-x-x^3-x^5-x^7-x^9-x^11+O(x^max(1,n-2))),n-3)  \\ _M. F. Hasler_, Jun 19 2011
%Y A191869 Cf. A000044.
%K A191869 nonn,easy
%O A191869 1,5
%A A191869 _Vladimir Joseph Stephan Orlovsky_, Jun 18 2011

%I A225396
%S A225396 1,1,2,3,5,8,13,21,34,55,88,142,229,369,595,959,1546,2492,4017,6475,
%T A225396 10438,16826,27123,43722,70479,113611,183139,295217,475885,767119,
%U A225396 1236583,1993351,3213249,5179704,8349597,13459412,21696349,34974155,56377758,90880011
%N A225396 Expansion of 1/(1 - x - x^2 + x^10 - x^12).
%C A225396 Limiting ratio is 1.61198..., the largest real root of -1 + x^2 - x^10 - x^11 + x^12 = 0.
%H A225396 Vincenzo Librandi, <a href="/A225396/b225396.txt">Table of n, a(n) for n = 0..1000</a>
%H A225396 Roger L. Bagula, <a href="/A225396/a225396.txt">Demo file</a>
%H A225396 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1).
%F A225396 a(0)=1, a(1)=1, a(2)=2, a(3)=3, a(4)=5, a(5)=8, a(6)=13, a(7)=21, a(8)=34, a(9)=55, a(10)=88, a(11)=142, a(n)=a(n-1)+a(n-2)- a(n-10)+ a(n-12). - _Harvey P. Dale_, Apr 12 2014
%t A225396 CoefficientList[Series[1/(1 - x - x^2 + x^10 - x^12), {x, 0, 50}], x]
%t A225396 LinearRecurrence[{1,1,0,0,0,0,0,0,0,-1,0,1},{1,1,2,3,5,8,13,21,34,55,88,142},50] (* _Harvey P. Dale_, Apr 12 2014 *)
%Y A225396 Cf. A117791, A107293, A204631, A225393, A225394, A029826, A147660.
%K A225396 nonn,easy
%O A225396 0,3
%A A225396 _Roger L. Bagula_, May 06 2013

%I A236212
%S A236212 0,0,0,0,0,0,1,1,2,3,5,8,13,21,36,63,113,206,386,736,1433,2849,5773,
%T A236212 11919,25059,53613,116658,258032,579856,1323273,3065246,7204159,
%U A236212 17172291,41498712,101635485,252180415,633710357,1612310803,4151993262,10819115820
%N A236212 Floor of the value of Riemann's xi function at n.
%C A236212 On the interval [1, infinity), the xi function takes real values and is (strictly) increasing, so a(n) <= a(n+1) for n >= 1.
%C A236212 Same as floor of the value of the xi function at 1-n, because of the functional equation xi(1-s) = x(s).
%H A236212 J. Sondow and C. Dumitrescu, <a href="http://arxiv.org/abs/1005.1104">A monotonicity property of Riemann's xi function and a reformulation of the Riemann Hypothesis</a>, Period. Math. Hungar. 60 (2010), 37-40.
%H A236212 E. Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/Xi-Function.html">Xi Function</a>
%H A236212 Wikipedia, <a href="https://en.wikipedia.org/wiki/Riemann_Xi_function">Riemann Xi function</a>
%H A236212 <a href="/index/Z#zeta_function">Index entries for zeta function</a>
%F A236212 a(n) = [xi(n)] for n > 0.
%e A236212 xi(1) = 1/2, so a(1) = [0.5] = 0.
%e A236212 xi(8) = (4*Pi^4)/225 = 1.7317…, so a(8) = [1.7] = 1.
%t A236212 xi[ s_] := If[ s == 1, 1/2, (s/2)*(s - 1)*Pi^(-s/2)*Gamma[ s/2]*Zeta[ s]]; Table[ Floor[ xi[ n]], {n, 40}]
%Y A236212 Cf. A002410.
%K A236212 nonn
%O A236212 1,9
%A A236212 _Jonathan Sondow_, Jan 25 2014

%I A248740
%S A248740 0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,597,584,181,765,946,
%T A248740 711,657,368,25,393,418,811,229,40,269,309,578,887,465,352,817,169,
%U A248740 986,155,141,296,437,733,170,903,73,976,49,25,74,99,173,272
%N A248740 a(n) = Fibonacci(n) mod 1000.
%C A248740 The sequence is periodic with period 1500 = A001175(1000).
%C A248740 A number m of {0, 1, ..., 999} is not in the range of this sequence, iff m is congruent to 4 or 6 mod 8.
%C A248740 These numbers are the 250 = 1000 - A066853(1000) numbers of the set {4, 6, 12, 14, ..., 996, 998}. E.g., a Fibonacci number will never end in the digits '100'.
%H A248740 Alois P. Heinz, <a href="/A248740/b248740.txt">Table of n, a(n) for n = 0..3000</a>
%H A248740 <a href="/index/Rec#order_1500">Index entries for linear recurrences with constant coefficients</a>, order 1500.
%F A248740 a(n) = (a(n-1) + a(n-2)) mod 1000 for n>1, a(0) = 0, a(1) = 1.
%e A248740 a(17) = (a(16) + a(15)) mod 1000 = (987 + 610) mod 1000 = 1597 mod 1000 = 597.
%p A248740 a:= proc(n) option remember;
%p A248740       `if`(n<2, n, irem(a(n-1)+a(n-2), 1000))
%p A248740     end:
%p A248740 seq(a(n), n=0..60);  # _Alois P. Heinz_, Oct 18 2015
%o A248740 (MAGMA) [Fibonacci(n) mod 1000: n in [0..80]]; // _Vincenzo Librandi_, Oct 17 2014
%o A248740 (PARI) vector(100,n,fibonacci(n-1)%1000) \\ _Derek Orr_, Oct 17 2014
%Y A248740 Cf. A000045, A003893, A105471.
%K A248740 nonn
%O A248740 0,4
%A A248740 _Franz Vrabec_, Oct 13 2014
%E A248740 More terms from _Vincenzo Librandi_, Oct 17 2014

# Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE

# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/

Search: seq:1,1,2,3,5,8,13,21
Showing 71-80 of 92

%I A274162
%S A274162 1,1,1,2,3,5,8,13,21,34,55,89,144,234,379,615,997,1617,2622,4252,6895,
%T A274162 11181
%N A274162 Number of real integers in n-th generation of tree T(3i) defined in Comments.
%C A274162 Let T* be the infinite tree with root 0 generated by these rules: if p is in T*, then p+1 is in T* and x*p is in T*. Let g(n) be the set of nodes in the n-th generation, so that g(0) = {0}, g(1) = {1}, g(2) = {2,x}, g(3) = {3,2x,x+1,x^2}, etc. Let T(r) be the tree obtained by substituting r for x.
%C A274162 See A274142 for a guide to related sequences.
%e A274162 If r = 3i, then g(3) = {3,2r,r+1, r^2}, in which the number of real integers is a(3) = 2.
%t A274162 z = 22; t = Join[{{0}}, Expand[NestList[DeleteDuplicates[Flatten[Map[{# + 1, x*#} &, #], 1]] &, {1}, z]]];
%t A274162 u = Table[t[[k]] /. x -> 3 I, {k, 1, z}]; Table[Count[Map[IntegerQ, u[[k]]], True], {k, 1, z}]
%Y A274162 Cf. A274142.
%K A274162 nonn,more
%O A274162 0,4
%A A274162 _Clark Kimberling_, Jun 12 2016
%E A274162 a(19)-a(21) from _Ryan Hitchman_, Sep 13 2017

%I A274163
%S A274163 1,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6766,
%T A274163 10948,17716,28667,46388,75063
%N A274163 Number of real integers in n-th generation of tree T(4i) defined in Comments.
%C A274163 Let T* be the infinite tree with root 0 generated by these rules:  if p is in T*, then p+1 is in T* and x*p is in T*.  Let g(n) be the set of nodes in the n-th generation, so that g(0) = {0}, g(1) = {1}, g(2) = {2,x}, g(3) = {3,2x,x+1,x^2}, etc.  Let T(r) be the tree obtained by substituting r for x.
%C A274163 For each integer k > 0, let s(k,n) be the number of integers in the n-th generation of T(k*i).  Conjecture:  there is a limiting sequence S(n) as k increases, and S(n) = F(n) for n >= 1, where F = A000045 (Fibonacci numbers).
%C A274163 From _Charlie Neder_, Jul 11 2018: (Start)
%C A274163 Assume for the moment that a complex number cannot be transformed back into an integer. If this is the case, then the real integers in g(n) are the real integers in g(n-1) plus 1 and the imaginary integers in g(n-1) times k*i, which are themselves k*i times the real integers in g(n-2), and so S(n) = S(n-1) + S(n-2) and S(n) = F(n).
%C A274163 However, the above assumption is false, but the earliest time such a transformation can take place is at g(k^2+5), following this path: 0 -> 1 -> k*i -> 1+k*i -> -k^2+k*i -> -(k^2-1)+k*i -> ... -> k*i -> -k^2.
%C A274163 Therefore s(k,n) matches the Fibonacci sequence for n < k^2+5 and S(n) = F(n). (End)
%C A274163 a(n) = A000045(n) only for 0 < n < 21. - _Robert G. Wilson v_, Jul 23 2018
%e A274163 If r = 4i, then g(3) = {3,2r,r+1, r^2}, in which the number of real integers is a(3) = 2.
%t A274163 z = 18; t = Join[{{0}}, Expand[NestList[DeleteDuplicates[Flatten[Map[{# + 1, x*#} &, #], 1]] &, {1}, z]]];
%t A274163 u = Table[t[[k]] /. x -> 4 I, {k, 1, z}]; Table[Count[Map[IntegerQ, u[[k]]], True], {k, 1, z}]
%Y A274163 See A274142 for a guide to related sequences.
%K A274163 nonn,more
%O A274163 0,4
%A A274163 _Clark Kimberling_, Jun 12 2016
%E A274163 a(21)-a(25) from _Robert G. Wilson v_, Jul 23 2018

%I A024466
%S A024466 1,0,0,1,1,2,3,0,0,1,1,2,3,5,8,13,21,34,55,1,1,2,3,5,8,13,21,34,55,89,
%T A024466 144,233,377,610,988,1598,2586,4184,6770,8,13,21,34,55,89,144,233,377,
%U A024466 610,987,1597,2584,4181,6765,10946,17712,28658,46370,75028,121398,196426
%N A024466 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)t(n+1-k), where k = [ (n+1)/2 ], s = (Fibonacci numbers), t = A023533.
%K A024466 nonn
%O A024466 1,6
%A A024466 _Clark Kimberling_

%I A025086
%S A025086 0,0,1,1,2,0,0,0,1,1,2,3,5,8,13,21,34,0,1,1,2,3,5,8,13,21,34,55,89,
%T A025086 144,233,377,610,988,1598,2586,4184,5,8,13,21,34,55,89,144,233,377,
%U A025086 610,987,1597,2584,4181,6765,10946,17712,28658,46370,75028,121398,196426,317824,514250
%N A025086 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)t(n-k+1), where k = [ n/2 ], s = (Fibonacci numbers), t = A023533.
%K A025086 nonn
%O A025086 1,5
%A A025086 _Clark Kimberling_

%I A067517
%S A067517 0,1,1,2,3,5,8,13,21,55,233,102334155
%N A067517 Fibonacci numbers which can be partitioned into group of digits which are Fibonacci numbers.
%K A067517 base,nonn
%O A067517 0,4
%A A067517 _Amarnath Murthy_, Feb 14 2002
%E A067517 102334155 from Larry Reeves (larryr(AT)acm.org), Nov 17 2003. Next term, if it exists, is > 2*10^18.

%I A069041
%S A069041 1,1,2,3,5,8,13,21,34,55,89,144,233,377,17711
%N A069041 Fibonacci numbers with at most two distinct digits.
%C A069041 Next term, if it exists, is > Fibonacci(2289000). - Lars Blomberg, May 10 2011
%t A069041 Fibonacci[ Select[Range[5*10^3], Length[Union[IntegerDigits[Fibonacci[ # ]], {}]] <= 2 &]]
%t A069041 Select[Fibonacci[Range[100]],Count[DigitCount[#],0]>7&] (* _Harvey P. Dale_, May 27 2018 *)
%K A069041 nonn,base
%O A069041 1,3
%A A069041 _Joseph L. Pe_, Apr 03 2002

%I A080787
%S A080787 1,1,2,3,5,8,13,21,24,25,29,34,43,47,50,57,57,64,71,75,76,81,87,88,95,
%T A080787 103,108,111,119,120,129,129,138,147,155,162,167,169,176,185,191,196,
%U A080787 197,203,210,213,213,216,219,225,234,239,243,252,255,257,262,269,271
%N A080787 a(1)=a(2)=1; a(n) = a(n-1) + last decimal digit of a(n-2).
%F A080787 a(n)=a(n-1)+a(n-2)(mod 10); for n>=3 a(n)-a(n-1)=A003893(n-2)=A000045(n-2)(mod 10)
%K A080787 base,nonn
%O A080787 1,3
%A A080787 _Benoit Cloitre_, Mar 12 2003

%I A093093
%S A093093 1,1,2,3,5,8,13,21,3,4,2,4,7,6,6,11,13,1,2,17,2,4,1,4,3,19,19,6,5,5,7,
%T A093093 2,2,3,8,25,11,1,0,1,2,9,4,5,11,33,3,6,1,2,1,1,3,11,13,9,1,6,4,4,3,6,
%U A093093 9,7,3,3,2,4,1,4,2,4,2,2,1,0,7,1,0,8,7,9,15,1,6,1,0,6,5,6,5,5,6,6,6,4,3,1,7
%N A093093 "Fibonacci in digits - up and down": start with a(1)=1, a(2)=1; repeatedly adjoin either the sum of the two previous terms (if that sum happens to be odd) or else adjoin digits of the sum of previous two terms (if that sum happens to be even).
%e A093093 ... a(8)=a(6)+a(7), a(9)=left digit of (a(7)+a(8)=13+21=3 4) as 34 is even, a(10)=right digit of (a(7)+a(8)=13+21=3 4) as 34 is even, a(13)=a(9)+a(10) as odd, ...
%Y A093093 Cf. A093086 to A093092.
%K A093093 nonn,base
%O A093093 1,3
%A A093093 _Bodo Zinser_, Mar 20 2004

%I A109609
%S A109609 1,1,2,3,5,8,13,21,34,55,89,144,234,378,612,990,1602,2592,4194,6786,
%T A109609 10980,17766,28746,46512,75259,121771,197030,318801,515831,834632,
%U A109609 1350463,2185095,3535558,5720653,9256211,14976864,24233076,39209940
%N A109609 Expansion of 1/((x-1)*(x+1)*(x^2+x+1)*(x^2+x-1)*(x^2-x+1)*(x^2+1)*(x^4-x^2+1)).
%C A109609 FAMP Code for s batch of sequences satisfying the recurrence relation as (a(n)): A*B with A = - .25'i - .25i' - .25'ii' + .25'jj' + .25'kk' + .25'jk' + .25'kj' - .25e, B = + 'i + i' + 'ji' + 'ki' + e. Sumtype is set to: sum[Y[15]]
%H A109609 <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, -1).
%F A109609 a(0)=1, a(1)=1, a(2)=2, a(3)=3, a(4)=5, a(5)=8, a(6)=13, a(7)=21, a(8)=34, a(9)=55, a(10)=89, a(11)=144, a(12)=234, a(13)=378, a(n)=a(n-1)+ a(n-2)+ a(n-12)-a(n-13)-a(n-14). - _Harvey P. Dale_, Sep 20 2013
%p A109609 seriestolist(series(1/((x-1)*(x+1)*(x^2+x+1)*(x^2+x-1)*(x^2-x+1)*(x^2+1)*(x^4-x^2+1)), x=0,40));
%t A109609 CoefficientList[Series[1/((x-1)(x+1)(x^2+x+1)(x^2+x-1)(x^2-x+1)(x^2+1)(x^4-x^2+1)),{x,0,40}],x] (* or *) LinearRecurrence[ {1,1,0,0,0,0,0,0,0,0,0,1,-1,-1},{1,1,2,3,5,8,13,21,34,55,89,144,234,378},40] (* _Harvey P. Dale_, Sep 20 2013 *)
%Y A109609 Cf. A000045.
%K A109609 nonn
%O A109609 0,3
%A A109609 _Creighton Dement_, Jul 31 2005

%I A117760
%S A117760 1,1,1,2,3,5,8,13,21,33,53,85,136,218,349,559,895,1433,2295,3675,5885,
%T A117760 9424,15091,24166,38698,61969,99234,158908,254467,407490,652533,
%U A117760 1044932,1673299,2679533,4290863,6871162,11003117,17619812,28215439,45182718
%N A117760 Expansion of 1/(1 - x - x^3 - x^5 - x^7).
%C A117760 Number of compositions of n into parts 1, 3, 5, and 7. - _David Neil McGrath_, Aug 18 2014
%H A117760 Milan Janjic, <a href="http://www.emis.ams.org/journals/JIS/VOL19/Janjic/janjic73.html">Binomial Coefficients and Enumeration of Restricted Words</a>, Journal of Integer Sequences, 2016, Vol 19, #16.7.3
%F A117760 a(n)=a(n-1)+a(n-3)+a(n-5)+a(n-7); a(0)=0, a(1)=1, a(2)=1, a(3)=1, a(4)=2, a(5)=3, a(6)=5.
%p A117760 a[0]:=0:a[1]:=1:a[2]:=1:a[3]:=1:a[4]:=2:a[5]:=3:a[6]:=5:for n from 7 to 45 do a[n]:=a[n-1]+a[n-3]+a[n-5]+a[n-7] od: seq(a[n],n=0..45);
%t A117760 CoefficientList[ Series[1/(1 - x - x^3 - x^5 - x^7), {x, 0, 40}], x]
%o A117760 (PARI) Vec( 1/(1-x-x^3-x^5-x^7)+O(x^66) ) \\ _Joerg Arndt_, Aug 19 2014
%K A117760 nonn
%O A117760 0,4
%A A117760 _Roger L. Bagula_, Apr 14 2006
%E A117760 Edited and extended by _N. J. A. Sloane_, Apr 20 2006

# Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE

# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/

Search: seq:1,1,2,3,5,8,13,21
Showing 81-90 of 92

%I A118627
%S A118627 1,1,2,3,5,8,13,21,6,30,24,55,31,87,56,144,88,233,145,379,234,614,380,
%T A118627 995,615,1611,996,2608,1612,4221,2609,6831,4222,11054,6832,17887,
%U A118627 11055,28943,17888,46832,28944,75777,46833,122611,75778,198390,122612,321003
%N A118627 a(1) = a(2) = 1. For n >=3, a(n) = the a(n-2)th integer, among those positive integers which are missing from the first (m-1) terms of the sequence, below a(n-1) if such a positive integer exists. Otherwise, a(n) = the a(n-2)th integer, among those positive integers which are missing from the first (m-1) terms of the sequence, above a(n-1).
%e A118627 The first 8 terms of the sequence are 1,1,2,3,5,8,13,21. Those integers which are missing from the first 8 terms of the sequence form the sequence 4,6,7,9,10,11,12,14,15,16,17,18,19,20,.. Counting down from a(8)=21 a total of a(7)=13 positions in this sequence of missing terms, we land on 6. So a(9) = 6.
%e A118627 There are fewer than a(8)=21 missing positive integers below a(9)=6, so we count UP to get a(10). a(10) is what we land on when counting up from 6 a total of a(8)=21 positions, skipping over terms which occur earlier in the sequence. So a(10) = 30.
%K A118627 easy,nonn
%O A118627 1,3
%A A118627 _Leroy Quet_, May 09 2006
%E A118627 More terms from _Joshua Zucker_, Jul 27 2006

%I A120659
%S A120659 0,1,1,2,3,5,8,13,21,34,55,2483,6053383,36651957891448,
%T A120659 1343366154248994863013047386,
%U A120659 1804632624381764689171354018874685160689875147803208124
%N A120659 A determinant sum sequence of the D5 dihehral 2 X 2 representation.
%D A120659 (*http : // mathworld.wolfram.com/DihedralGroupD3.html*)
%F A120659 << MathWorld`Groups` M0 = DihedralGroupMatrices[5]; s[n_] := M0[[n]] a[0] = Table[Fibonacci[n], {n, 0, 9}]; a[1] = Table[Fibonacci[n], {n,1, 10}]; a[n_] := a[n] = {a[n - 1][[2]], a[n - 1][[3]], a[n - 1][[4]], a[n - 1][[5]], a[n - 1][[6]], a[n - 1][[7]], a[n - 1][[8]], a[n - 1][[9]], a[n - 1][[10]], Abs[Det[Sum[a[n - 1][[i]]*s[i], {i, 1, 10}]]]}
%t A120659 Clear[a, f, s] (*http : // mathworld.wolfram.com/DihedralGroupD3.html*) << MathWorld`Groups` M0 = DihedralGroupMatrices[5]; s[n_] := M0[[n]] a[0] = Table[Fibonacci[n], {n, 0, 9}]; a[1] = Table[Fibonacci[n], {n, 1, 10}]; a[n_] := a[n] = {a[n - 1][[2]], a[n - 1][[3]], a[n - 1][[4]], a[n - 1][[5]], a[n - 1][[6]], a[n - 1][[7]], a[n - 1][[8]], a[n - 1][[9]], a[n - 1][[10]], Abs[Det[Sum[a[n - 1][[i]]*s[i], {i, 1, 10}]]]} Table[Floor[a[n][[1]]], {n, 0, 16}]
%Y A120659 Cf. A120495, A120496.
%K A120659 nonn,uned
%O A120659 0,4
%A A120659 _Roger L. Bagula_, Aug 10 2006

%I A141450
%S A141450 1,1,2,1,1,4,1,1,3,7,1,1,2,4,12,1,1,2,4,8,19,1,1,2,3,6,11,30,1,1,2,3,
%T A141450 6,9,19,45,1,1,2,3,5,8,15,26,67,1,1,2,3,5,8,13,21,41,97,1,1,2,3,5,7,
%U A141450 12,18,31,56,139,1,1,2,3,5,7,12,17,28,45,83,195,1,1,2,3,5,7,11,16,25,38,63
%N A141450 Upper right triangle of the number of m's in all partitions of n.
%C A141450 The "last" column read from the bottom is A000041.
%C A141450 Mirror of triangle A066633. - _Omar E. Pol_, May 01 2012
%e A141450 A000070: 1, 2, 4, 7, 12, 19, 30, 45, 67, 97, 139, 195, 272, 373, 508, ...,
%e A141450 A024786: 0, 1, 1, 3, 4, 8, 11, 19, 26, 41, 56, 83, 112, 160, 213, ...,
%e A141450 A024787: 0, 0, 1, 1, 2, 4, 6, 9, 15, 21, 31, 45, 63, 87, 122, ...,
%e A141450 A024788: 0, 0, 0, 1, 1, 2, 3, 6, 8, 13, 18, 28, 38, 55, 74, ...,
%e A141450 A024789: 0, 0, 0, 0, 1, 1, 2, 3, 5, 8, 12, 17, 25, 35, 50, ...,
%e A141450 A024790: 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 12, 16, 24, 33, ...,
%e A141450 A024791: 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 16, 23, ...,
%e A141450 A024792: 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, ...,
%e A141450 A024793: 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, ...,
%e A141450 A024794: 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, ...
%t A141450 (* First do ) Needs["Combinatorica`"] (* then *) f[n_, m_] := Count[Flatten@ Partitions@ n, m]; Table[ f[n, m], {n, 13}, {m, n, 1, -1}]
%Y A141450 Cf. A000041, A000070, A024786, A024787, A024788, A024789, A024790, A024791, A024792, A024793, A024794.
%K A141450 nonn,tabl
%O A141450 1,3
%A A141450 _Robert G. Wilson v_, Aug 07 2008

%I A175712
%S A175712 0,1,1,2,3,5,8,13,21,33,54,79,133,176,309,365,674,709,1383,1300,2683,
%T A175712 2267,4950,3785,8735,6085,14820,9465,24285,14302,38587
%N A175712 The third column of the Lucas Fibonacci sum of binomials A175685.
%F A175712 a(n) = A175685(n,3), n>=1 .
%t A175712 Table[Sum[Binomial[n - j - 1, j], {j, Floor[(n - 1)/2] - 3, Floor[(
%t A175712     n - 1)/2]}], {n, 0, 30}]
%Y A175712 Cf. A000045, A175685, A175686.
%K A175712 nonn
%O A175712 0,4
%A A175712 _Roger L. Bagula_, Dec 04 2010

%I A177376
%S A177376 1,1,2,3,5,8,13,21,34,55,144,233,377,610,2584,4181,6765,17711,28657,
%T A177376 46368,75025,317811,832040,3524578,5702887,24157817,102334155,
%U A177376 165580141,701408733,20365011074,86267571272,225851433717,17167680177565
%N A177376 Fibonacci numbers whose decimal expansion does not contain any digit 9.
%C A177376 The probability that Fib(n) contains no 9's goes to zero as n grows to infinity. It appears that the largest term is F(188). [Corrected by _Jon E. Schoenfield_, May 08 2010]
%e A177376 a(11)=144 since it is the 11th Fibonacci containing no 9's
%t A177376 Select[Fibonacci[Range[100]],DigitCount[#,10,9]==0&] (* _Harvey P. Dale_, Jan 22 2014 *)
%Y A177376 Cf. A000045, A177194, A177195, A177231, A177245, A177246, A176253, A177247, A177272, A177372, A177374
%K A177376 nonn,base
%O A177376 1,3
%A A177376 _Carmine Suriano_, May 07 2010

%I A178526
%S A178526 1,1,0,1,1,1,1,1,2,1,1,1,2,3,2,1,1,2,3,5,3,1,1,2,3,5,8,5,1,1,2,3,5,8,
%T A178526 13,8,1,1,2,3,5,8,13,21,13,1,1,2,3,5,8,13,21,34,21,1,1,2,3,5,8,13,21,
%U A178526 34,55,34,1,1,2,3,5,8,13,21,34,55,89,55,1,1,2,3,5,8,13,21,34,55,89,144,89
%N A178526 Triangle read by rows: T(n,k) is the number of nodes of cost k in the Fibonacci tree of order n.
%C A178526 A Fibonacci tree of order n (n>=2) is a complete binary tree whose left subtree is the Fibonacci tree of order n-1 and whose right subtree is the Fibonacci tree of order n-2; each of the Fibonacci trees of order 0 and 1 is defined as a single node. In a Fibonacci tree the cost of a left (right) edge is defined to be 1 (2). The cost of a node of a Fibonacci tree is defined to be the sum of the costs of the edges that form the path from the root to this node.
%C A178526 The sum of the entries in row n is A001595(n) = 2F(n+1) - 1, where F(m)=A000045(m) (the Fibonacci numbers).
%C A178526 Sum(k*T(n,k), 0<=k<=n)=A178525(n).
%C A178526 _Daniel Forgues_, Aug 10 2012: (Start)
%C A178526 The falling diagonals are, starting from the rightmost one, with index 0:
%C A178526   d_0(i) = F(i-1), i >= 0;
%C A178526   d_j(i) = F(i+1), j >= 1, i >= 0.
%C A178526 Equivalently, as a single expression:
%C A178526   d_j(i) = F(i+1-2*0^j), j >= 0, i >= 0. (End)
%D A178526 D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417.
%H A178526 Y. Horibe, <a href="http://www.fq.math.ca/Scanned/20-2/horibe.pdf">An entropy view of Fibonacci trees</a>, Fibonacci Quarterly, 20, No. 2, 1982, 168-178.
%F A178526 T(n,k)=F(k+1) if k<n; T(n,n)=F(n-1); T(n,k)=0 if k>n; here F(m)=A000045(m) (the Fibonacci numbers).
%F A178526 G.f.: (1-tz+tz^2)/[(1-z)(1-tz-t^2*z^2)].
%F A178526 The enumerating polynomials P[n] of row n are given by P[0]=1, P[n]=P[n-1]+F(n-1)*(t^{n-1}+t^n) for n>=1, where F(m)=A000045(m) (the Fibonacci numbers).
%e A178526 In the Fibonacci tree /\ of order 2 we have a node of cost 0 (the root), a node of cost 1 (the left leaf), and a node of cost 2 (the right leaf).
%e A178526 Triangle starts:
%e A178526 1;
%e A178526 1,0;
%e A178526 1,1,1;
%e A178526 1,1,2,1;
%e A178526 1,1,2,3,2;
%e A178526 1,1,2,3,5,3;
%e A178526 1,1,2,3,5,8,5;
%p A178526 with(combinat): T := proc (n, k) if k < n then fibonacci(k+1) elif k = n then fibonacci(n-1) else 0 end if end proc: for n from 0 to 12 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form
%Y A178526 Cf. A000045, A001595, A178525.
%K A178526 nonn,tabl
%O A178526 0,9
%A A178526 _Emeric Deutsch_, Jun 16 2010

%I A218032
%S A218032 1,1,1,1,2,3,5,8,13,21,35,57,94,154,254,417,687,1129,1859,3057,5032,
%T A218032 8277,13623,22412,36883,60684,99862,164312,270384,444899,732093,
%U A218032 1204629,1982228,3261701,5367131,8831505,14532200,23912499,39347839,64746320,106539481,175309363,288469809
%N A218032 G.f. A(x) satisfies A(x) = 1 + x / (1- x * A(x^2)).
%C A218032 What does this sequence count?
%o A218032 (PARI)
%o A218032 N=66;  R=O('x^N);  x='x+R;
%o A218032 F = 1 + x;
%o A218032 for (k=1,N+1, F = 1 + x / (1- x * subst(F,'x,'x^2) ) + R; );
%o A218032 Vec(F)
%K A218032 nonn
%O A218032 0,5
%A A218032 _Joerg Arndt_, Oct 19 2012

%I A280198
%S A280198 1,1,1,2,3,5,8,13,21,33,53,86,138,222,357,574,923,1484,2387,3839,6173,
%T A280198 9927,15964,25672,41284,66389,106762,171686,276091,443989,713988,
%U A280198 1148179,1846411,2969252,4774918,7678647,12348195,19857396,31933099,51352294,82580715,132799801,213558181,343427445,552272966,888121883,1428207656
%N A280198 Expansion of 1/(1 - Sum_{k>=1} mu(2*k-1)^2*x^(2*k-1)), where mu() is the Moebius function (A008683).
%C A280198 Number of compositions (ordered partitions) into odd squarefree parts (A056911).
%H A280198 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Squarefree.html">Squarefree</a>
%H A280198 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F A280198 G.f.: 1/(1 - Sum_{k>=1} mu(2*k-1)^2*x^(2*k-1)).
%e A280198 a(4) = 3 because we have [3, 1], [1, 3] and [1, 1, 1, 1].
%t A280198 nmax = 46; CoefficientList[Series[1/(1 - Sum[MoebiusMu[2 k - 1]^2 x^(2 k - 1), {k, 1, nmax}]), {x, 0, nmax}], x]
%Y A280198 Cf. A005117, A008683, A056911, A134345, A280194.
%K A280198 nonn
%O A280198 0,4
%A A280198 _Ilya Gutkovskiy_, Dec 28 2016

%I A293865
%S A293865 0,0,0,1,1,2,3,5,8,13,21,37,56,90,144,239,374,592,948,1558,2431,3848,
%T A293865 6127,9972,15602,24658,39158,63265,99110,156505,248040,398675,625024,
%U A293865 986241,1560763,2498832,3919561,6180914,9770162,15594972,24470070,38567903,60907330
%N A293865 Number of self-intersecting walks of length n on a square lattice such that at each point the angle turns 90 degrees.
%C A293865 It is assumed that the first walk turns left and that all walks end when they intersect themselves.
%H A293865 MathStackExchange, <a href="https://math.stackexchange.com/questions/2471391/expected-number-of-steps-before-intersection">Expected Number of Steps Before Intersection</a>, Oct 2017.
%F A293865 For n>2, a(n) = 2*A189722(n-1) - A189722(n). - _Jens Randrup Rasmussen_, Oct 29 2017
%e A293865 For n = 4 we have the simplest self-intersecting walk, which is a square.
%e A293865 For n = 5 we have the walk:
%e A293865 (0,0), (0,1), (-1,1), (-1, 2), (0,2), (0,1)
%e A293865 For n = 6 we have the walks:
%e A293865 (0,0), (0,1), (-1,1), (-1, 0), (-2,0), (-2,1), (-1,1)
%e A293865 (0,0), (0,1), (-1,1), (-1, 2), (-2,2), (-2,1), (-1,1)
%o A293865 (Visual Basic for Excel)
%o A293865 Const N = 50
%o A293865 Const MaxSteps = 43
%o A293865 Dim BeenHere() As Boolean
%o A293865 Dim LoopBacks(MaxSteps) As Long
%o A293865 Dim PosX As Integer, PosY As Integer
%o A293865 Sub Macro1()
%o A293865   ReDim BeenHere(N, N)
%o A293865   PosX = N / 2: PosY = N / 2
%o A293865   BeenHere(PosX, PosY) = True
%o A293865   PosX = PosX + 1
%o A293865   BeenHere(PosX, PosY) = True
%o A293865   PosY = PosY - 1
%o A293865   BeenHere(PosX, PosY) = True
%o A293865   DoSteps 2, 3, PosX, PosY, BeenHere()
%o A293865   For i = 4 To MaxSteps
%o A293865     Cells(i - 1, 3).Value = i
%o A293865     Cells(i - 1, 4).Value = LoopBacks(i)
%o A293865   Next i
%o A293865 End Sub
%o A293865 Sub DoSteps(ByVal StepNo As Integer, Dir As Integer, X As Integer, Y As Integer, BH() As Boolean)
%o A293865 Dim BH2() As Boolean
%o A293865 Dim X1 As Integer, Y1 As Integer, X2 As Integer, Y2 As Integer
%o A293865 Dim Dir1 As Integer, Dir2 As Integer
%o A293865   BH2 = BH
%o A293865   StepNo = StepNo + 1
%o A293865   Select Case Dir
%o A293865   Case 1, 3 ' North or South
%o A293865       Dir1 = 2: X1 = X + 1: Y1 = Y
%o A293865       Dir2 = 4: X2 = X - 1: Y2 = Y
%o A293865   Case 2, 4 ' East or West
%o A293865       Dir1 = 1: Y1 = Y + 1: X1 = X
%o A293865       Dir2 = 3: Y2 = Y - 1: X2 = X
%o A293865   End Select
%o A293865   If BH2(X1, Y1) Then
%o A293865     LoopBacks(StepNo) = LoopBacks(StepNo) + 1
%o A293865   ElseIf StepNo < MaxSteps Then
%o A293865     BH2(X1, Y1) = True
%o A293865     DoSteps StepNo, Dir1, X1, Y1, BH2()
%o A293865     BH2(X1, Y1) = False
%o A293865   End If
%o A293865   If BH2(X2, Y2) Then
%o A293865     LoopBacks(StepNo) = LoopBacks(StepNo) + 1
%o A293865   ElseIf StepNo < MaxSteps Then
%o A293865     BH2(X2, Y2) = True
%o A293865     DoSteps StepNo, Dir2, X2, Y2, BH2()
%o A293865   End If
%o A293865 End Sub
%Y A293865 This sequence gives the number of self-intersecting walks while A189722 gives the number of self-avoiding walks.
%K A293865 nonn,walk
%O A293865 1,6
%A A293865 _Jens Randrup Rasmussen_, Oct 18 2017
%E A293865 The terms starting from a(11) and the program corrected by _Jens Randrup Rasmussen_, Oct 29 2017

%I A039834
%S A039834 1,1,0,1,-1,2,-3,5,-8,13,-21,34,-55,89,-144,233,-377,610,-987,1597,
%T A039834 -2584,4181,-6765,10946,-17711,28657,-46368,75025,-121393,196418,
%U A039834 -317811,514229,-832040,1346269,-2178309,3524578,-5702887,9227465,-14930352,24157817
%N A039834 a(n+2) = -a(n+1) + a(n) (signed Fibonacci numbers) with a(-2) = a(-1) = 1; or Fibonacci numbers (A000045) extended to negative indices.
%C A039834 Knuth defines the negaFibonacci numbers as follows: F(-1) = 1, F(-2) = -1, F(-3) = 2, F(-4) = -3, F(-5) = 5, ..., F(-n) = (-1)^(n-1) F(n). See A215022, A215023 for the negaFibonacci representation of n. - _N. J. A. Sloane_, Aug 03 2012
%C A039834 The ratio of successive terms converges to -1/phi. - _Jonathan Vos Post_, Dec 10 2006
%C A039834 Let a(n) := F(n) * (-1)^binom(n, 2). Then a(m - n) * a(m + n) = a(m + 1) * a(m - 1) * a(n)^2 - a(n + 1) * a(n - 1) * a(m)^2. This plus gcd(f[n], f[m]) = |f[gcd(n, m)]| makes a[] a strong elliptic divisibility sequence. Likewise F(n) * (-1)^binom(n - 1, 2), but no other asSIGNation (mod scaling). - _Bill Gosper_, May 28 2008
%C A039834 The sequence a(n), n >= 0 := 0, 1, -1, 2, -3, 5, -8, 13, ... is the inverse binomial transform of A000045. - _Philippe Deléham_, Oct 28 2008
%C A039834 Equals the INVERTi transform of A038754, assuming that an additional A038754(0) = 1 is added in front of A038754, and that the a(n) are prefixed with another 1 and then get offset 0. - _Gary W. Adamson_, Jan 08 2011
%C A039834 If we remove a(-2) and then set the offset to 0, we have the INVERT transform of a signed A011782: (1, -1, 2, -4, 8, -16, 32, ...).- _Gary W. Adamson_, Jan 08 2011
%C A039834 The sequence 0, 1, -1, 2, -3, 5, -8, 13, -21, 34, -55, 89, -144,.. (starting at offset 0) is the Lucas U(-1,-1) sequence. - _R. J. Mathar_, Jan 08 2013
%C A039834 This sequence appears in the formula for 1/rho(5)^n, with rho(5) = (1 + sqrt(5))/2 = phi (golden section), when written in the power basis <1, rho(5)> of the quadratic number field Q(rho(5)): 1/rho(5)^n = a(n+1) * 1 + a(n) * rho(5), n >= -2. - _Wolfdieter Lang_, Nov 04 2013
%C A039834 a(n) = A227431(n + 4, n + 3). - _Reinhard Zumkeller_, Feb 01 2014
%D A039834 D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.3, p. 168, Eq. (145).
%D A039834 D. Shtefan and I. Dobrovolska, The sums of the consecutive Fibonacci numbers, Fib. Q., 56 (2018), 229-236.
%H A039834 Indranil Ghosh, <a href="/A039834/b039834.txt">Table of n, a(n) for n = -2..4773</a> (terms -2..500 from T. D. Noe)
%H A039834 M. Cetin Firengiz, A. Dil, <a href="http://www.nntdm.net/papers/nntdm-20/NNTDM-20-4-21-32.pdf">Generalized Euler-Seidel method for second order recurrence relations</a>, Notes on Number Theory and Discrete Mathematics, Vol. 20, 2014, No. 4, 21-32.
%H A039834 Jiřı Jina and Pavel Trojovský, <a href="http://dx.doi.org/10.12732/ijpam.v88i4.11">On determinants of some tridiagonal matrices connected with Fibonacci numbers</a>, International Journal of Pure and Applied Mathematics, Volume 88 No. 4 2013, 569-575; ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version).
%H A039834 J. Pan, <a href="https://cs.uwaterloo.ca/journals/JIS/OL13/Pan/pan8.html">Multiple Binomial Transforms and Families of Integer Sequences </a>, J. Int. Seq. 13 (2010), 10.4.2
%H A039834 J. Pan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Pan/pan12.html"> Some Properties of the Multiple Binomial Transform and the Hankel Transform of Shifted Sequences </a>, J. Int. Seq. 14 (2011) # 11.3.4, remark 14.
%H A039834 Wikipedia, <a href="http://en.wikipedia.org/wiki/Lucas_sequence#Specific_names">Lucas sequence</a>
%H A039834 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-1,1)
%H A039834 <a href="/index/Lu#Lucas">Index entries for Lucas sequences</a>
%F A039834 G.f.: (1+2*x)/(x^2*(1+x-x^2)).
%F A039834 a(n-2) = Sum_{k, 0 <= k <= n}(-2)^k*A055830(n, k). - _Philippe Deléham_, Oct 18 2006
%F A039834 a(n) = ((phi - 1)^n + 1/phi*(-(1/phi) - 1)^(n+1))/sqrt(5), where phi = (1 + sqrt(5))/2. - _Arkadiusz Wesolowski_, Oct 28 2012
%F A039834 a(n) = sum(k = 1..n, binomial(n - 1, k - 1)*fib(k)*(-1)^(n - k)), n > 0, fib(k) = A000045(k), a(0) = 1. - _Perminova Maria_, Jan 22 2013
%F A039834 G.f.: 1 + x/( Q(0) - x ) where Q(k) = 1 - x/(x*k - 1 )/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Feb 23 2013
%F A039834 G.f.: 2 - 2/(Q(0) + 1) where Q(k) = 1 + 2*x/(1 - x/(x + 1/Q(k+1) )); (continued fraction ). - _Sergei N. Gladkovskii_, Apr 05 2013
%F A039834 G.f.: 1 + x^2 + x^3 + x/Q(0) , where Q(k)= 1 + (k+1)*x/(1 - x/(x + (k+1)/Q(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Apr 23 2013
%F A039834 G.f.: 1/(G(0)*x^3) + (2*x^2+x-1)/x^3, where G(k) = 1 + 2*x*(k+1)/(k + 2 - x*(k+2)*(k+3)/(x*(k+3) + (k+1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 27 2013
%F A039834 G.f.: Q(0)/x - 1/x + 1+ x, where Q(k) = 1 + x^2 + x^3 + k*x*(1+x^2) - x^2*(1 + x*(k+2))*(1+k*x)/Q(k+1) ; (continued fraction). - _Sergei N. Gladkovskii_, Jan 13 2014
%F A039834 a(n) = -(-1)^n*A000045(n), at least for all n >= 0 (and also for n < 0 if A000045 is extended to negative indices). - _M. F. Hasler_, May 10 2017
%e A039834 From - _Wolfdieter Lang_, Nov 04 2013 (Start)
%e A039834 With the golden section phi = rho(5) = (1 + sqrt(5))/2:
%e A039834 n = -2: phi^2 = a(-1)*1 + a(-2)*phi = 1 + phi,
%e A039834 n = -1: phi = a(0)*1 + a(-1)*phi = phi, (trivial)
%e A039834 n =  0: 1/phi^0 =  a(1)*1 + a(0)*phi = 1, (trivial)
%e A039834 n =  1: 1/phi = a(2)*1 + a(1)*phi = -1 + phi,
%e A039834 n =  2: 1/phi^2 = a(3)*1 + a(2)*phi = 2 - phi, ... (End)
%e A039834 G.f. = x^-2 + x^-1 + x - x^2 + 2*x^3 - 3*x^4 + 5*x^5 - 8*x^6 + 13*x^7 -...
%p A039834 a:= n-> (Matrix([[0, 1], [1, -1]])^n) [1,2]: seq(a(n), n=-2..50); # _Alois P. Heinz_, Nov 01 2008
%t A039834 LinearRecurrence[{-1, 1}, {1, 1}, 60] (* _Vladimir Joseph Stephan Orlovsky_, May 25 2011 *)
%t A039834 Fibonacci[-Range[-2, 37]] (* _Michael Somos_, Jun 04 2016 *)
%o A039834 (PARI) a(n) = fibonacci(-n);
%o A039834 (Haskell)
%o A039834 a039834 n = a039834_list !! (n+2)
%o A039834 a039834_list = 1 : 1 : zipWith (-) a039834_list (tail a039834_list)
%o A039834 -- _Reinhard Zumkeller_, Jul 05 2013
%o A039834 (Sage)
%o A039834 def A039834():
%o A039834     x, y = 1, 1
%o A039834     while true:
%o A039834         yield x
%o A039834         x, y = y, x - y
%o A039834 a = A039834()
%o A039834 [a.next() for i in range(40)]  # _Peter Luschny_, Jul 11 2013
%o A039834 (Sage)
%o A039834 def A039834_list(len):
%o A039834     R.<t> = LaurentSeriesRing(ZZ, 't', default_prec = len)
%o A039834     f = (-2*t-1)/(t^4-t^3-t^2)
%o A039834     return f.list()
%o A039834 A039834_list(40) # _Peter Luschny_, Nov 21 2014
%Y A039834 Cf. A000045, A038754, A011782, A215022, A215023.
%K A039834 sign,easy,nice
%O A039834 -2,6
%A A039834 Alexander Grasser (pyropunk(AT)usa.net)
%E A039834 Signs corrected by _Len Smiley_ and _N. J. A. Sloane_

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# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/

Search: seq:1,1,2,3,5,8,13,21
Showing 91-92 of 92

%I A152163
%S A152163 1,-1,0,-1,-1,-2,-3,-5,-8,-13,-21,-34,-55,-89,-144,-233,-377,-610,
%T A152163 -987,-1597,-2584,-4181,-6765,-10946,-17711,-28657,-46368,-75025,
%U A152163 -121393,-196418,-317811,-514229,-832040,-1346269,-2178309,-3524578,-5702887
%N A152163 a(n)=a(n-1)+a(n-2), n>1 ; a(0)=1, a(1)=-1 .
%H A152163 Vincenzo Librandi, <a href="/A152163/b152163.txt">Table of n, a(n) for n = 0..1000</a>
%H A152163 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1).
%F A152163 G.f.: (1-2*x)/(1-x-x^2).
%F A152163 a(n) = Sum_{k, 0<=k<=n}A147703(n,k)*(-2)^k.
%F A152163 a(n) = -Fibonacci(n-2) for n >= 2, and for all n if A000045 is extended in the natural way to negative indices; see also A039834. [Extended by _M. F. Hasler_, May 10 2017]
%F A152163 a(n) = (1/2)*{[(1/2)+(1/2)*sqrt(5)]^n+[(1/2)-(1/2)*sqrt(5)]^n}+(3/10)*sqrt(5)*{[(1/2)-(1/2) *sqrt(5)]^n-[(1/2)+(1/2)*sqrt(5)]^n}, with n>=0 [From _Paolo P. Lava_, Dec 01 2008]
%F A152163 a(n) = (-1)^n*A039834(n-2). - _R. J. Mathar_, Mar 22 2011
%F A152163 G.f.: (1/(1-Q(0))-1)*(1-2*x)/x where Q(k)=1 - x^k/(1 - x/( x - x^k/Q(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Feb 23 2013
%F A152163 G.f.: 2 - 2/(Q(0)+1) where Q(k) = 1 - 2*x/(1 - x/(x - 1/Q(k+1) )); (continued fraction ). - _Sergei N. Gladkovskii_, Apr 05 2013
%F A152163 a(n) = A000045(n+1)-2*A000045(n). - _R. J. Mathar_, Jun 26 2013
%F A152163 G.f.: 1 - x - x^3*Q(0)/2, where Q(k) = 1 + 1/(1 - x*(6*k+1 + x)/(x*(6*k+4 + x) + 1/Q(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jan 02 2014
%F A152163 G.f.: 1+1/x - x - Q(0)/x, where Q(k) = 1 + x^2 - x^3 - k*x*(1+x^2) - x^2*( x*(k+2)-1)*( k*x -1)/Q(k+1) ; (continued fraction). - _Sergei N. Gladkovskii_, Jan 13 2014
%t A152163 LinearRecurrence[{1,1},{1,-1},40] (* _Harvey P. Dale_, Oct 09 2012 *)
%o A152163 (MAGMA) I:=[1, -1]; [n le 2 select I[n] else Self(n-1)+Self(n-2): n in [1..40]]; // _Vincenzo Librandi_, Feb 23 2013
%o A152163 (PARI) a(n)=-fibonacci(n-2) \\ _M. F. Hasler_, May 10 2017
%Y A152163 Cf. A000045.
%K A152163 easy,sign
%O A152163 0,6
%A A152163 _Philippe Deléham_, Nov 27 2008

%I A236191
%S A236191 0,1,1,2,-3,-5,-8,13,21,34,-55,-89,-144,233,377,610,-987,-1597,-2584,
%T A236191 4181,6765,10946,-17711,-28657,-46368,75025,121393,196418,-317811,
%U A236191 -514229,-832040,1346269,2178309,3524578,-5702887,-9227465,-14930352,24157817,39088169
%N A236191 (-1)^floor( (n-1) / 3 ) * F(n), where F = Fibonacci.
%H A236191 Vincenzo Librandi, <a href="/A236191/b236191.txt">Table of n, a(n) for n = 0..1000</a>
%H A236191 <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,-4,0,0,1).
%F A236191 G.f.: (x + x^2 + 2*x^3 + x^4 - x^5) / (1 + 4*x^3 - x^6).
%F A236191 a(n) = -4 * a(n-3) + a(n-6). a(-n) = (-1)^floor( (n-2) / 3) * a(n) for all n in Z.
%F A236191 a(n) * a(n+3) = a(n+1)^2 - a(n+2)^2 for all n in Z.
%e A236191 G.f. = x + x^2 + 2*x^3 - 3*x^4 - 5*x^5 - 8*x^6 + 13*x^7 + 21*x^8 + ...
%t A236191 a[ n_] := (-1)^Quotient[n-1, 3] Fibonacci[n];
%t A236191 CoefficientList[Series[(x + x^2 + 2 x^3 + x^4 - x^5)/(1 + 4 x^3 - x^6), {x, 0, 50}], x] (* _Vincenzo Librandi_, Jan 20 2014 *)
%o A236191 (PARI) {a(n) = (-1)^( (n-1) \ 3) * fibonacci( n)};
%o A236191 (MAGMA) I:=[0,1,1,2,-3,-5]; [n le 6 select I[n] else -4*Self(n-3)+Self(n-6): n in [1..40]]; // _Vincenzo Librandi_, Jan 20 2014
%Y A236191 Cf. A000045.
%K A236191 sign,easy
%O A236191 0,4
%A A236191 _Michael Somos_, Jan 19 2014

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