A099623 a(n) = Sum_{k=0..floor(n/2)} C(n-k,k+2)*2^(n-k-2)*(3/2)^k.

0, 0, 1, 6, 27, 104, 369, 1242, 4039, 12828, 40077, 123758, 379011, 1153872, 3498025, 10572354, 31884543, 96010436, 288788613, 867967830, 2607282235, 7828953720, 23501774241, 70536546986, 211674885687, 635160738924, 1905765565309

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

• Index entries for linear recurrences with constant coefficients, signature (6,-9,-4,12).

Programs

Mathematica

LinearRecurrence[{6,-9,-4,12},{0,0,1,6},30] (* Harvey P. Dale, Mar 27 2016 *)

Formula

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

a(n) = 6*a(n-1)-9*a(n-2)-4*a(n-3)+12*a(n-4).

a(n) = -(n/3+7/9)*2^n +(-1)^n/36 +3^(n+1)/4. - R. J. Mathar, Dec 16 2024