A387692 a(n) = Sum_{k=0..floor(n/3)} 2^(n-3*k) * binomial(2*n-4*k+1,2*k).

1, 2, 4, 11, 36, 116, 357, 1078, 3256, 9879, 30040, 91352, 277673, 843802, 2564124, 7792163, 23680412, 71965100, 218701901, 664632878, 2019811632, 6138187055, 18653894640, 56689020016, 172277420241, 523549505522, 1591062148084, 4835223303739, 14694199419988

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1500

Index entries for linear recurrences with constant coefficients, signature (4,-4,2,4,0,-1).

Formula

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

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

Mathematica

CoefficientList[Series[(1-2*x+x^3)/((1-2*x+x^3)^2 - 4*x^3), {x, 0, 28}], x] (* Stefano Spezia, Sep 06 2025 *)
Table[Sum[2^(n-3*k)*Binomial[2*n-4k+1, 2*k], {k, 0, Floor[n/3]}], {n, 0, 40}] (* Vincenzo Librandi, Sep 07 2025 *)

Prog

(PARI) a(n) = sum(k=0, n\3, 2^(n-3*k) * binomial(2*n-4*k+1, 2*k));
(Magma) [&+[2^(n-3*k)* Binomial(2*n-4*k+1, 2*k): k in [0..Floor (n/3)]]: n in [0..40]]; // Vincenzo Librandi, Sep 07 2025

Crossrefs

Cf. A001653, A099488, A387693.

Cf. A387695.

Sequence in context: A219861 A193058 A179379 * A086611 A035098 A328425

Adjacent sequences: A387689 A387690 A387691 * A387693 A387694 A387695

Keyword

nonn,easy

Author

Seiichi Manyama, Sep 06 2025

Status

approved