A015584 Expansion of g.f. x/(1 - 9*x - 8*x^2).

0, 1, 9, 89, 873, 8569, 84105, 825497, 8102313, 79524793, 780541641, 7661073113, 75193991145, 738034505209, 7243862476041, 71099038326041, 697842244742697, 6849372509292601, 67227090541574985, 659838794948515673, 6476365878869240937, 63566003269411293817

Offset: 0

Author

Olivier Gérard

Keywords

• nonn
• easy

Comments

Pisano period lengths: 1, 1, 4, 1, 24, 4, 6, 1, 4, 24, 10, 4, 12, 6, 24, 1,144, 4, 15, 24, ... . - R. J. Mathar, Aug 10 2012

For n >= 2, the number of positive integers with n-1 decimal digits in which adjacent digits differ by at most 8. - Edwin Hermann, Apr 19 2025

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (9,8).

Crossrefs

Cf. A015579, A099371.

Programs

Magma

[n le 2 select n-1 else 9*Self(n-1) + 8*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 15 2012

Mathematica

LinearRecurrence[{9, 8}, {0, 1}, 30] (* Vincenzo Librandi, Nov 15 2012 *)
CoefficientList[Series[x/(1-9x-8x^2),{x,0,30}],x] (* Harvey P. Dale, Sep 06 2022 *)

PARI

concat(0, Vec(x / (1-9*x-8*x^2) + O(x^30))) \\ Colin Barker, May 16 2017

Sage

[lucas_number1(n,9,-8) for n in range(0, 19)] # Zerinvary Lajos, Apr 26 2009

Formula

a(n) = 9*a(n-1) + 8*a(n-2).

a(n) = (-((9-sqrt(113))/2)^n + ((9+sqrt(113))/2)^n) / sqrt(113). - Colin Barker, May 16 2017

E.g.f.: 2*exp(9*x/2)*sinh(sqrt(113)*x/2)/sqrt(113). - Stefano Spezia, Oct 25 2023

Extensions

Extended by T. D. Noe, May 23 2011