A368529 a(n) = Sum_{k=1..n} k^2 * 4^(n-k).

0, 1, 8, 41, 180, 745, 3016, 12113, 48516, 194145, 776680, 3106841, 12427508, 49710201, 198841000, 795364225, 3181457156, 12725828913, 50903315976, 203613264265, 814453057460, 3257812230281, 13031248921608, 52124995686961, 208499982748420, 833999930994305

Offset: 0

Author

Seiichi Manyama, Dec 28 2023

Keywords

• nonn
• easy

Links

• Paolo Xausa, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (7,-15,13,-4).

Crossrefs

Cf. A000290, A000330, A047520, A368528.

Cf. A052161, A368524.

Cf. A014825, A368530.

Programs

Mathematica

LinearRecurrence[{7, -15, 13, -4}, {0, 1, 8, 41}, 30] (* Paolo Xausa, Jan 29 2024 *)

PARI

a(n) = sum(k=1, n, k^2*4^(n-k));

Formula

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

a(n) = 7*a(n-1) - 15*a(n-2) + 13*a(n-3) - 4*a(n-4).

a(n) = A052161(n-1) + A052161(n-2) for n > 1.

a(n) = (5*4^(n+1) - (9*n^2 + 24*n + 20))/27.

a(0) = 0; a(n) = 4*a(n-1) + n^2.