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

Author

Olivier Gérard

Keywords

• nonn
• easy

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)

Crossrefs

Row m=8 of A135597.

Programs

Magma

[n le 2 select 1 else 8*Self(n-1) + Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 08 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 *)

PARI

x='x+O('x^30); Vec((1-7*x)/(1-8*x-x^2)) \\ G. C. Greubel, Dec 19 2017

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