A387602 a(n) = (1/2) * Sum_{k=0..floor(n/3)} 2^(n-2*k) * binomial(2*n-4*k+2,2*k+1).

1, 4, 12, 36, 120, 416, 1420, 4768, 15968, 53664, 180736, 608640, 2048336, 6891968, 23191104, 78044352, 262644608, 883866624, 2974400960, 10009502720, 33684265984, 113355412480, 381467226112, 1283724873728, 4320028764416, 14537889756160, 48923344206848

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 (4,-4,4,8,0,-4).

Crossrefs

Cf. A375278, A387604.

Cf. A387510.

Programs

Magma

[&+[2^(n-2*k)* Binomial(2*n-4*k+2, 2*k+1)/2: k in [0..Floor (n/3)]]: n in [0..35]]; // Vincenzo Librandi, Sep 03 2025

Mathematica

Table[Sum[2^(n-2*k)*Binomial[2*n-4*k+2, 2*k+1]/2,{k,0,Floor[n/3]}],{n,0,40}] (* Vincenzo Librandi, Sep 03 2025 *)

PARI

a(n) = sum(k=0, n\3, 2^(n-2*k)*binomial(2*n-4*k+2, 2*k+1))/2;

Formula

G.f.: B(x)^2, where B(x) is the g.f. of A387510.

G.f.: 1/((1-2*x-2*x^3)^2 - 16*x^4).

a(n) = 4*a(n-1) - 4*a(n-2) + 4*a(n-3) + 8*a(n-4) - 4*a(n-6).