A087451 G.f.: (2-x)/((1+2*x)*(1-3*x)); e.g.f.: exp(3*x) + exp(-2*x); a(n) = 3^n + (-2)^n.

2, 1, 13, 19, 97, 211, 793, 2059, 6817, 19171, 60073, 175099, 535537, 1586131, 4799353, 14316139, 43112257, 129009091, 387682633, 1161737179, 3487832977, 10458256051, 31385253913, 94134790219, 282446313697, 847255055011, 2541932937193, 7625463267259, 22877060890417

Offset

0,1

Comments

Generalized Lucas-Jacobsthal numbers.

Pisano period lengths: 1, 1, 1, 2, 4, 1, 6, 2, 3, 4, 5, 2, 12, 6, 4, 4, 16, 3, 18, 4,... - R. J. Mathar, Aug 10 2012

Links

Bruce Nye, Table of n, a(n) for n = 0..1000

Drexel University, Generation X and Y

G. Everest, Y. Puri and T. Ward, Integer sequences counting periodic points, arXiv:math/0204173 [math.NT], 2002.

OEIS Wiki, Autosequence

Index entries for linear recurrences with constant coefficients, signature (1,6)

Formula

a(0) = 2, a(1) = 1, a(n) = a(n-1) + 6*a(n-2).

The sequence 1, 13, 19,... is a(n+1) = 3*3^n - 2*(-2)^n.

exp(Sum_{n >= 1} a(n)*x^n/n) = Sum_{n >= 0} A015441(n+1)*x^n. - Peter Bala, Mar 30 2015

a(n) = 2*A015441(n+1) - A015441(n), a formula given by Paul Curtz for autosequences of the 2nd kind. - Jean-François Alcover, Jun 02 2017

Maple

seq(3^n + (-2)^n, n=0..28);

Mathematica

a[0] = 2; a[1] = 1; a[n_] := a[n] = a[n - 1] + 6a[n - 2]; a /@ Range[0, 28] (* Robert G. Wilson v, Feb 02 2006 *)

Prog

(SageMath) [lucas_number2(n, 1, -6) for n in range(0, 28)] # Zerinvary Lajos, Apr 30 2009
(PARI) Vec((2-x)/((1+2*x)*(1-3*x)) + O(x^29)) \\ Bruce Nye, Apr 29 2026

Crossrefs

Cf. A014551, A087452, A015441.

Sequence in context: A032001 A358205 A118679 * A063558 A174170 A264373

Adjacent sequences: A087448 A087449 A087450 * A087452 A087453 A087454

Keyword

easy,nonn

Author

Paul Barry, Sep 06 2003

Status

approved