A387693 a(n) = Sum_{k=0..floor(n/4)} 2^(n-4*k) * binomial(2*n-6*k+1,2*k).

1, 2, 4, 8, 19, 52, 148, 416, 1141, 3078, 8248, 22096, 59319, 159560, 429624, 1157024, 3115481, 8387146, 22575900, 60765464, 163557995, 440246780, 1185025900, 3189793984, 8586134701, 23111701262, 62210729264, 167455038112, 450745122735, 1213288033040

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,-4,0,2,4,0,0,-1).

Formula

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

a(n) = 4*a(n-1) - 4*a(n-2) + 2*a(n-4) + 4*a(n-5) - a(n-8).

Mathematica

CoefficientList[Series[(1-2*x+x^4)/((1-2*x+x^4)^2 - 4*x^4), {x, 0, 29}], x] (* Stefano Spezia, Sep 06 2025 *)
Table[Sum[2^(n-4*k)*Binomial[2*n-6k+1, 2*k], {k, 0, Floor[n/4]}], {n, 0, 40}] (* Vincenzo Librandi, Sep 07 2025 *)

Prog

(PARI) a(n) = sum(k=0, n\4, 2^(n-4*k) * binomial(2*n-6*k+1, 2*k));
(Magma) [&+[2^(n-4*k)* Binomial(2*n-6*k+1, 2*k): k in [0..Floor (n/4)]]: n in [0..40]]; // Vincenzo Librandi, Sep 07 2025

Crossrefs

Cf. A001653, A099488, A387692.

Cf. A387691, A387696.

Sequence in context: A100133 A099598 A269023 * A173310 A320178 A128816

Adjacent sequences: A387690 A387691 A387692 * A387694 A387695 A387696

Keyword

nonn,easy

Author

Seiichi Manyama, Sep 06 2025

Status

approved