A387690 a(n) = Sum_{k=0..floor(n/3)} 2^(n-3*k) * binomial(2*n-4*k,2*k).

1, 2, 4, 9, 28, 92, 289, 878, 2648, 8017, 24360, 74088, 225249, 684554, 2080188, 6321369, 19210468, 58380900, 177419969, 539178134, 1638556144, 4979554209, 15132817936, 45988498832, 139758636609, 424725225682, 1290736069556, 3922535089833, 11920548260972

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 - 8*x^4).

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 - 8*x^4), {x, 0, 28}], x] (* Stefano Spezia, Sep 06 2025 *)
(* Alternative: *)
Table[Sum[2^(n-3*k)*Binomial[2*n-4k, 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, 2*k));
(Magma) [&+[2^(n-3*k)* Binomial(2*n-4*k, 2*k): k in [0..Floor (n/3)]]: n in [0..40]]; // Vincenzo Librandi, Sep 07 2025

Crossrefs

Cf. A001541, A387689, A387691.

Sequence in context: A176068 A296264 A343166 * A219665 A375632 A241404

Adjacent sequences: A387687 A387688 A387689 * A387691 A387692 A387693

Keyword

nonn,easy

Author

Seiichi Manyama, Sep 06 2025

Status

approved