A099625 a(n) = Sum_{k=0..floor(n/2)} C(n-k,k+2)*2^(n-k-2)*(1/2)^k.

0, 0, 1, 6, 25, 88, 281, 842, 2413, 6692, 18101, 48014, 125393, 323376, 825393, 2088850, 5248853, 13110844, 32584653, 80639446, 198844281, 488813768, 1198491913, 2931934938, 7158830781, 17450923092, 42480107365, 103283553054, 250859152801, 608759955040, 1476163691105

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) = 3*u*a(n-1)-(3*u^2-v)*a(n-2)+(u^3-2*u*v)*a(n-3)+u^2*v*a(n-4).

Links

• Paolo Xausa, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (6,-11,4,4).

Crossrefs

Cf. A000129, A099623, A099624.

Programs

Mathematica

LinearRecurrence[{6, -11, 4, 4}, {0, 0, 1, 6}, 35] (* Paolo Xausa, Jan 15 2025 *)

Formula

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

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

a(n) = 6*a(n-1)-11*a(n-2)+4*a(n-3)+4*a(n-4).

a(n) = A000129(n+3) -(n+5)*2^n. - R. J. Mathar, Dec 16 2024