A015443 Generalized Fibonacci numbers: a(n) = a(n-1) + 8*a(n-2).

1, 1, 9, 17, 89, 225, 937, 2737, 10233, 32129, 113993, 371025, 1282969, 4251169, 14514921, 48524273, 164643641, 552837825, 1869986953, 6292689553, 21252585177, 71594101601, 241614783017, 814367595825, 2747285859961

Offset

0,3

Comments

Construct a graph as follows: form the graph whose adjacency matrix is the tensor product of that of P_3 and [1,1;1,1], then add a loop at each of the extremity nodes. a(n-1) counts walks of length n between adjacent nodes. - Paul Barry, Nov 12 2004

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, 9*a(n-2) equals the number of 9-colored compositions of n with all parts >= 2, such that no adjacent parts have the same color. - Milan Janjic, Nov 26 2011

Pisano period lengths: 1, 1, 6, 1, 24, 6, 16, 1, 6, 24, 110, 6, 56, 16, 24, 2, 16, 6, 60, 24, ... - R. J. Mathar, Aug 10 2012

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Joerg Arndt, Matters Computational (The Fxtbook), p. 318

M. Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.

Index entries for linear recurrences with constant coefficients, signature (1,8)

Formula

a(n) = (((1+sqrt(33))/2)^(n+1) - ((1-sqrt(33))/2)^(n+1))/sqrt(33).

a(n) = Sum_{k=0..n} A109466(n,k)*(-8)^(n-k). - Philippe Deléham, Oct 26 2008

G.f.: 1/(1-x-8*x^2). - R. J. Mathar, Apr 07 2011

a(n) = (Sum_{1<=k<=n+1, k odd} C(n+1,k)*33^((k-1)/2))/2^n. - Vladimir Shevelev, Feb 05 2014

Mathematica

CoefficientList[Series[1/(1-x-8*x^2), {x, 0, 50}], x] (* G. C. Greubel, Apr 30 2017 *)

Prog

(Sage) [lucas_number1(n, 1, -8) for n in range(1, 27)] # Zerinvary Lajos, Apr 22 2009
(Magma) [ n eq 1 select 1 else n eq 2 select 1 else Self(n-1)+8*Self(n-2): n in [1..30] ]; // Vincenzo Librandi, Aug 23 2011
(PARI) a(n)=Vec(1/(1-x-8*x^2)+O(x^99)) \\ Charles R Greathouse IV, Feb 03 2014

Crossrefs

Cf. A015442, A015441, A100302, A100303.

Sequence in context: A166705 A116526 A197396 * A253212 A217965 A121442

Adjacent sequences: A015440 A015441 A015442 * A015444 A015445 A015446

Keyword

nonn,easy

Author

Olivier Gérard

Status

approved