A015454 Generalized Fibonacci numbers.

1, 1, 9, 73, 593, 4817, 39129, 317849, 2581921, 20973217, 170367657, 1383914473, 11241683441, 91317382001, 741780739449, 6025563297593, 48946287120193, 397595860259137, 3229713169193289, 26235301213805449, 213112122879636881

Offset

0,3

Comments

a(n)/a(n-1) tends to (8 + 2*sqrt(17))/2 = exp ArcSinh 4 = A176458. - Gary W. Adamson, Dec 26 2007

For n>=1, row sums of triangle for numbers 8^k*C(m,k) with duplicated diagonals. - Vladimir Shevelev, Apr 13 2012

For n>=1, a(n) equals the numbers of words of length n-1 on alphabet {0,1,...,8} containing no subwords ii, (i=0,1,...,7). - Milan Janjic, Jan 31 2015

a(n+1) is the number of nonary sequences of length n such that no two consecutive terms have distance 5. - David Nacin, May 31 2017

Links

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

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

Tanya Khovanova, Recursive Sequences

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

Formula

a(n) = 8*a(n-1) + a(n-2).

a(n) = Sum_{k=0..n} 7^k*A055830(n,k). - Philippe Deléham, Oct 18 2006

G.f.: (1-7*x)/(1-8*x-x^2). - Philippe Deléham, Nov 20 2008

For n>=2, a(n) = F_n(8)+F_(n+1)(8), where F_n(x) is Fibonacci polynomial (cf.A049310): F_n(x) = Sum_{i=0..floor((n-1)/2)} C(n-i-1,i)*x^(n-2*i-1). - Vladimir Shevelev, Apr 13 2012

a(n) = A041025(n) -7*A041025(n-1). - R. J. Mathar, Jul 06 2012

Mathematica

LinearRecurrence[{8, 1}, {1, 1}, 30] (* Vincenzo Librandi, Nov 08 2012 *)
CoefficientList[Series[(1-7*x)/(1-8*x-x^2), {x, 0, 50}], x] (* G. C. Greubel, Dec 19 2017 *)

Prog

(Magma) [n le 2 select 1 else 8*Self(n-1) + Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 08 2012
(PARI) x='x+O('x^30); Vec((1-7*x)/(1-8*x-x^2)) \\ G. C. Greubel, Dec 19 2017

Crossrefs

Row m=8 of A135597.

Sequence in context: A164588 A023001 A277672 * A343353 A121246 A365774

Adjacent sequences: A015451 A015452 A015453 * A015455 A015456 A015457

Keyword

nonn,easy

Author

Olivier Gérard

Status

approved