A368528 a(n) = Sum_{k=1..n} k^2 * 3^(n-k).

0, 1, 7, 30, 106, 343, 1065, 3244, 9796, 29469, 88507, 265642, 797070, 2391379, 7174333, 21523224, 64569928, 193710073, 581130543, 1743391990, 5230176370, 15690529551, 47071589137, 141214767940, 423644304396, 1270932913813, 3812798742115

Offset: 0

Author

Seiichi Manyama, Dec 28 2023

Keywords

• nonn
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (6,-12,10,-3).

Crossrefs

Cf. A000290, A000330, A047520, A368529.

Cf. A052150, A368524.

Programs

PARI

a(n) = sum(k=1, n, k^2*3^(n-k));

Formula

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

a(n) = 6*a(n-1) - 12*a(n-2) + 10*a(n-3) - 3*a(n-4).

a(n) = A052150(n-1) + A052150(n-2) for n > 1.

a(n) = (3^(n+1) - (n^2 + 3*n + 3))/2.

a(0) = 0; a(n) = 3*a(n-1) + n^2.