A099624 a(n) = Sum_{k=0..floor(n/2)} C(n-k,k+2)*3^(n-k-2)*(4/3)^k.

0, 0, 1, 9, 58, 318, 1591, 7503, 33976, 149436, 643261, 2724357, 11395654, 47210154, 194121811, 793526571, 3228811492, 13090123272, 52917410041, 213437246145, 859342367890, 3455021317590, 13875655896751, 55677180731079

Offset: 0

Author

Paul Barry, Oct 25 2004

Keywords

• easy
• nonn

Comments

In general a(n) = Sum_{k=0..floor(n/2)} C(n-k,k+2)*u^(n-k-2)*(v/u)^k has g.f. x^2/((1-u*x)^2(1-u*x-v*x^2)) and satisfies the recurrence a(n) = 3u*a(n-1)-(3u^2-v)*a(n-2)+(u^3-2uv)*a(n-3)+u^2^v*a(n-4).

Links

• Index entries for linear recurrences with constant coefficients, signature (9,-23,3,36).

Crossrefs

Cf. A099623.

Formula

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

a(n) = 9*a(n-1)-23*a(n-2)+3*a(n-3)+36*a(n-4).

a(n) = -(n/4+13/16)*3^n +(-1)^n/80 +4^(n+1)/5 . - R. J. Mathar, Dec 16 2024