A087451 G.f.: (2-x)/((1+2x)(1-3x)); e.g.f.: exp(3x)+exp(-2x); 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

Offset: 0

Author

Paul Barry, Sep 06 2003

Keywords

• easy
• nonn

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

• 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)

Crossrefs

Cf. A014551, A087452, A015441.

Programs

Maple

for i from 0 to 20 do print(3^i+(-2)^i) od; # Gary Detlefs, Dec 20 2009

Mathematica

a[0] = 2; a[1] = 1; a[n_] := a[n] = a[n - 1] + 6a[n - 2]; a /@ Range[0, 25] (* Robert G. Wilson v, Feb 02 2006 *)

Sage

[lucas_number2(n,1,-6) for n in range(0, 26)] # Zerinvary Lajos, Apr 30 2009

Formula

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

a(n) = 3^n + (-2)^n. - Gary Detlefs, Dec 20 2009

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