A100133 a(n) = Sum_{k=0..floor(n/4)} C(n-2k,2k) * 3^k * 2^(n-4k).

1, 2, 4, 8, 19, 50, 136, 368, 985, 2618, 6940, 18392, 48763, 129338, 343120, 910304, 2415025, 6406898, 16996852, 45090728, 119620579, 317340098, 841868632, 2233386320, 5924932489, 15718204970, 41698695820, 110622122360

Offset: 0

Author

Paul Barry, Nov 06 2004

Keywords

• easy
• nonn

Comments

Binomial transform of 1,1,1,1,4,4,10,10,28,28,76,... (g.f. (1-x)(1+x)^2/(1-2x^2-2x^4)).

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-4,0,3)

Crossrefs

Cf. A100131, A100132.

Programs

PARI

a(n) = sum(k=0, n\4, binomial(n-2*k,2*k) * 3^k * 2^(n-4*k)); \\ Michel Marcus, Oct 09 2021

PARI

my(p=Mod('x, 'x^4-4*'x^3+4*'x^2-3)); a(n) = subst(lift(p^n),'x,2); \\ Kevin Ryde, Feb 02 2023

Formula

G.f.: (1-2x)/((1-2x)^2-3x^4).

a(n) = 4*a(n-1) - 4*a(n-2) + 3*a(n-4). [corrected by Kevin Ryde, Feb 02 2023]