A099432 Convolution of A030195(n) (generalized (3,3)-Fibonacci) with itself.

1, 6, 33, 162, 756, 3402, 14931, 64314, 273051, 1145988, 4764744, 19656756, 80561061, 328316814, 1331513397, 5377120038, 21633427836, 86747114430, 346810621815, 1382826606210, 5500378861551, 21830478128136, 86469557676048

Offset: 0

Author

Paul Barry, Oct 15 2004

Keywords

• easy
• nonn

Links

• Index entries for linear recurrences with constant coefficients, signature (6,-3,-18,-9).

Crossrefs

Cf. A073388.

Programs

Mathematica

LinearRecurrence[{6,-3,-18,-9},{1,6,33,162},30] (* Harvey P. Dale, May 20 2011 *)

Formula

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

a(n) = 6*a(n-1) - 3*a(n-2) - 18*a(n-3) - 9*a(n-4). [corrected by Harvey P. Dale, May 20 2011]

a(n) = Sum_{k=0..floor((n+2)/2)} k*binomial(n-k+2, k)*3^(n-k+1).

a(n) = (sqrt(7)*n + 2*sqrt(7) - sqrt(3))*(5*sqrt(7)/98 + sqrt(3)/14)*(3*sqrt(21)/2 + 15/2)^(n/2) + (15/2 - 3*sqrt(21)/2)^(n/2)*(sqrt(7)*n + 2*sqrt(7) + sqrt(3))*(5*sqrt(7)/98 - sqrt(3)/14)*(-1)^n.