A387556 a(n) = (1/2) * Sum_{k=0..floor(n/3)} 2^(n-2*k) * binomial(2*k+2,2*n-6*k+1).

1, 0, 0, 4, 8, 0, 12, 80, 48, 32, 448, 896, 336, 1920, 8064, 7872, 8320, 50688, 101824, 79616, 262400, 879616, 1096704, 1490944, 5888256, 11923456, 13332480, 34886656, 100288512, 146227200, 228961280, 702910464, 1430450176, 1968660480, 4587044864

Offset: 0

Author

Seiichi Manyama, Sep 02 2025

Keywords

• nonn
• easy
• new

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1500
• Index entries for linear recurrences with constant coefficients, signature (0,0,4,8,0,-4,16,-16).

Crossrefs

Cf. A387485.

Programs

Magma

[&+[2^(n-2*k) * Binomial(2*k+2, 2*n-6*k+1)/2: k in [0..Floor(n/3)]]: n in [0..40]]; // Vincenzo Librandi, Sep 02 2025

Mathematica

Table[Sum[2^(n-2*k)*Binomial[2*k+2, 2*n-6*k+1]/2,{k,0,Floor[n/3]}],{n,0,40}] (* Vincenzo Librandi, Sep 02 2025 *)

PARI

a(n) = sum(k=0, n\3, 2^(n-2*k)*binomial(2*k+2, 2*n-6*k+1))/2;

Formula

G.f.: B(x)^2, where B(x) is the g.f. of A387485.

G.f.: 1/((1-2*x^3-4*x^4)^2 - 32*x^7).

a(n) = 4*a(n-3) + 8*a(n-4) - 4*a(n-6) + 16*a(n-7) - 16*a(n-8).