cp's OEIS Frontend

A015441 Generalized Fibonacci numbers.

%I A015441 #140 Jun 30 2025 23:46:04
%S A015441 0,1,1,7,13,55,133,463,1261,4039,11605,35839,105469,320503,953317,
%T A015441 2876335,8596237,25854247,77431669,232557151,697147165,2092490071,
%U A015441 6275373061,18830313487,56482551853,169464432775,508359743893,1525146340543,4575304803901,13726182847159
%N A015441 Generalized Fibonacci numbers.
%C A015441 a(n) is the coefficient of x^(n-1) in the bivariate Fibonacci polynomials F(n)(x,y) = xF(n-1)(x,y) + yF(n-2)(x,y), F(0)(x,y)=0, F(1)(x,y)=1, when y=6x^2. - Mario Catalani (mario.catalani(AT)unito.it), Dec 06 2002
%C A015441 For n>=1: number of length-(n-1) words with letters {0,1,2,3,4,5,6,7} where no two consecutive letters are nonzero, see fxtbook link below. - _Joerg Arndt_, Apr 08 2011
%C A015441 Starting with offset 1 and convolved with (1, 3, 3, 3, ...) = A003462: (1, 4, 13, 40, ...). - _Gary W. Adamson_, May 28 2009
%C A015441 a(n) is identical to its inverse binomial transform signed. Differences: A102901. - _Paul Curtz_, Feb 23 2010
%C A015441 The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n>=2, 7*a(n-2) equals the number of 7-colored compositions of n with all parts >=2, such that no adjacent parts have the same color. - _Milan Janjic_, Nov 26 2011
%C A015441 Pisano period lengths: 1, 1, 1, 2, 20, 1, 6, 2, 3, 20, 5, 2, 12, 6, 20, 4, 16, 3, 18, 20, ... - _R. J. Mathar_, Aug 10 2012
%C A015441 A015441 and A015518 are the only integer sequences (from the family of homogeneous linear recurrence relation of order 2 with positive integer coefficients with initial values a(0)=0 and a(1)=1) whose ratio a(n+1)/a(n) converges to 3 as n approaches infinity. - _Felix P. Muga II_, Mar 14 2014
%C A015441 This is an autosequence of the first kind: the array of successive differences shows a main diagonal of zeros and the inverse binomial transform is identical to the sequence (with alternating signs). - Pointed out by _Paul Curtz_, Dec 05 2016
%C A015441 First two upper diagonals: A000400(n).
%C A015441 This is a variation on the Starhex honeycomb configuration A332243, see illustration in links. It is an alternating pattern of the 2nd iteration of the centered hexagonal numbers A003215 and centered 12-gonal 'Star' numbers A003154. - _John Elias_, Oct 06 2021
%H A015441 G. C. Greubel, <a href="/A015441/b015441.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..200 from T. D. Noe)
%H A015441 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, p. 317-318
%H A015441 A. Abdurrahman, <a href="https://arxiv.org/abs/1909.10889">CM Method and Expansion of Numbers</a>, arXiv:1909.10889 [math.NT], 2019.
%H A015441 John Elias, <a href="/A015441/a015441.png">Illustration: A Starhex honeycomb variation</a>
%H A015441 Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Janjic/janjic63.html">On Linear Recurrence Equations Arising from Compositions of Positive Integers</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
%H A015441 H. Li and T. MacHenry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/MacHenry/machenry7.html">Permanents and Determinants, Weighted Isobaric Polynomials, and Integer Sequences</a>, J. Int. Seq. 16 (2013) #13.3.5, example 48.
%H A015441 F. P. Muga II, <a href="https://www.researchgate.net/publication/267327689_Extending_the_Golden_Ratio_and_the_Binet-de_Moivre_Formula">Extending the Golden Ratio and the Binet-de Moivre Formula</a>, March 2014; Preprint on ResearchGate.
%H A015441 A. G. Shannon and J. V. Leyendekkers, <a href="http://nntdm.net/volume-21-2015/number-2/35-42/">The Golden Ratio family and the Binet equation</a>, Notes on Number Theory and Discrete Mathematics, Vol. 21, No. 2, (2015), 35-42.
%H A015441 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,6).
%F A015441 G.f.: x/((1+2*x)*(1-3*x)).
%F A015441 a(n) = a(n-1) + 6*a(n-2).
%F A015441 a(n) = (1/5)*((3^n)-((-2)^n)). - henryk.wicke(AT)stud.uni-hannover.de
%F A015441 E.g.f.: (exp(3*x) - exp(-2*x))/5. - _Paul Barry_, Apr 20 2003
%F A015441 a(n+1) = Sum_{k=0..ceiling(n/2)} 6^k*binomial(n-k, k). - _Benoit Cloitre_, Mar 06 2004
%F A015441 a(n) = (A000244(n) - A001045(n+1)(-1)^n - A001045(n)(-1)^n)/5. - _Paul Barry_, Apr 27 2004
%F A015441 The binomial transform of [1,1,7,13,55,133,463,...] is A122117. - _Philippe Deléham_, Oct 19 2006
%F A015441 a(n+1) = Sum_{k=0..n} A109466(n,k)*(-6)^(n-k). - _Philippe Deléham_, Oct 26 2008
%F A015441 a(n) = 3a(n-1) + (-1)^(n+1)*A000079(n-1). - _Paul Curtz_, Feb 23 2010
%F A015441 G.f.: Q(0) -1, where Q(k) = 1 + 6*x^2 + (k+2)*x - x*(k+1 + 6*x)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Oct 06 2013
%F A015441 a(n) = (Sum_{1<=k<=n, k odd} binomial(n,k)*5^(k-1))/2^(n-1). - _Vladimir Shevelev_, Feb 05 2014
%F A015441 a(-n) = -(-1)^n * a(n) / 6^n for all n in Z. - _Michael Somos_, Mar 18 2014
%F A015441 From _Peter Bala_, Apr 01 2015: (Start)
%F A015441 Sum_{n >= 0} a(n+1)*x^n = exp( Sum_{n >= 1} A087451(n)*x^n/n ).
%F A015441 For k = 0, 1, 2, ... and for n >= 1, (5^k)*a(n) | a((5^k)*n).
%F A015441 The expansion of exp( Sum_{n >= 1} a(5*n)/(5*a(n))*x^n/n ) has integral coefficients. Cf. A001656. (End)
%F A015441 From  _Peter Bala_, Jun 27 2025: (Start)
%F A015441 Sum_{n >= 1} (-6)^n/(a(n)*a(n+1)) = -2, since (-6)^n/(a(n)*a(n+1)) = (-2)^n/a(n) - (-2)^(n+1)/a(n+1) for n >= 1.
%F A015441 The following are examples of telescoping infinite products:
%F A015441 Product_{n >= 0} (1 + 6^n/a(2*n+2)) = 6, since (1 + 6^(2*n-1)/a(4*n))*(1 + 6^(2*n)/a(4*n+2)) = (6 - 4^(n+1)/b(n)) / (6 - 4^n/b(n-1)), where b(n) = (2*4^n + 3*9^n)/5 = A096951(n). Similarly,
%F A015441 Product_{n >= 1} (1 - 6^n/a(2*n+2)) = 3/13.
%F A015441 Product_{n >= 0} (1 + (-6)^n/a(2*n+2)) = 6/5.
%F A015441 Product_{n >= 1} (1 - (-6)^n/a(2*n+2)) = 15/13.
%F A015441 exp( Sum_{n >= 1} a(2*n)/a(n)*x^n/n ) = Sum_{n >= 0} a(n+1)*x^n. (End)
%e A015441 G.f. = x + x^2 + 7*x^3 + 13*x^4 + 55*x^5 + 133*x^6 + 463*x^7 + 1261*x^8 + ...
%p A015441 A015441:=n->(1/5)*((3^n)-((-2)^n)); seq(A015441(n), n=0..30); # _Wesley Ivan Hurt_, Mar 14 2014
%t A015441 a[n_]:=(MatrixPower[{{1,4},{1,-2}},n].{{1},{1}})[[2,1]]; Table[Abs[a[n]], {n,-1,40}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 19 2010 *)
%t A015441 LinearRecurrence[{1,6},{0,1},30] (* _Harvey P. Dale_, Apr 26 2011 *)
%t A015441 CoefficientList[Series[x/((1 + 2 x) (1 - 3 x)), {x, 0, 29}], x] (* _Michael De Vlieger_, Dec 05 2016 *)
%o A015441 (PARI) {a(n) = (3^n - (-2)^n) / 5};
%o A015441 (Sage) [lucas_number1(n,1,-6) for n in range(0, 27)] # _Zerinvary Lajos_, Apr 22 2009
%o A015441 (Magma) I:=[0,1]; [n le 2 select I[n] else Self(n-1) + 6*Self(n-2): n in [1..30]]; // _G. C. Greubel_, Jan 24 2018
%Y A015441 Cf. A000079, A003462, A016153, A109466, A122117. Cf. A001656, A087451. Cf. A000400.
%K A015441 nonn,easy,nice
%O A015441 0,4
%A A015441 _Olivier Gérard_