A015535 Expansion of x/(1 - 5*x - 2*x^2).

0, 1, 5, 27, 145, 779, 4185, 22483, 120785, 648891, 3486025, 18727907, 100611585, 540513739, 2903791865, 15599986803, 83807517745, 450237562331, 2418802847145, 12994489360387, 69810052496225, 375039241201899, 2014816311001945, 10824160037413523, 58150432809071505

Offset

0,3

Comments

Pisano period lengths: 1, 1, 3, 2, 8, 3, 48, 2, 3, 8, 110, 6, 168, 48, 24, 4, 8, 3, 45, 8, ... - R. J. Mathar, Aug 10 2012

This is the Lucas sequence U(5,-2). - Bruno Berselli, Jan 08 2013

For n > 0, a(n) equals the number of words of length n-1 over {0,1,...,6} in which 0 and 1 avoid runs of odd lengths. - Milan Janjic, Jan 08 2017

Links

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

Michel Mollard, Munarini graphs: a generalization of Fibonacci cubes and Pell graphs. Part I, arXiv:2605.14613 [math.CO], 2026. See p. 17 (Prop. 4.8).

Wikipedia, Lucas sequence: Specific names.

Index entries for linear recurrences with constant coefficients, signature (5,2).

Formula

a(n) = 5*a(n-1) + 2*a(n-2) with n > 1, a(0)=0, a(1)=1.

From Stefano Spezia, Sep 07 2025: (Start)

a(n) = (((5 + sqrt(33))/2)^n - ((5 - sqrt(33))/2)^n)/sqrt(33).

E.g.f.: 2*exp(5*x/2)*sinh(sqrt(33)*x/2)/sqrt(33). (End)

Mathematica

LinearRecurrence[{5, 2}, {0, 1}, 30] (* Vincenzo Librandi, Nov 12 2012 *)

Prog

(SageMath) [lucas_number1(n, 5, -2) for n in range(0, 22)] # Zerinvary Lajos, Apr 24 2009
(Magma) [n le 2 select n-1 else 5*Self(n-1)+2*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 12 2012
(PARI) my(x='x+O('x^30)); concat([0], Vec(x/(1-5*x-2*x^2))) \\ G. C. Greubel, Jan 01 2018

Crossrefs

Cf. A201002 (prime subsequence).

Sequence in context: A052225 A293295 A343208 * A026292 A100193 A158869

Adjacent sequences: A015532 A015533 A015534 * A015536 A015537 A015538

Keyword

nonn,easy,changed

Author

Olivier Gérard

Status

approved