A387695 a(n) = Sum_{k=0..floor(n/3)} 2^(n-3*k) * binomial(2*n-4*k+1,2*k+1).

1, 6, 20, 57, 164, 492, 1505, 4602, 14008, 42545, 129208, 392584, 1193121, 3626142, 11020076, 33489769, 101774332, 309290388, 939930945, 2856443826, 8680709552, 26380596577, 80170385200, 243637058512, 740410793665, 2250101653494, 6838038387588, 20780736161177

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).

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), {x, 0, 27}], x] (* Stefano Spezia, Sep 06 2025 *)
Table[Sum[2^(n-3*k)*Binomial[2*n-4k+1, 2*k+1], {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+1));
(Magma) [&+[2^(n-3*k)* Binomial(2*n-4*k+1, 2*k+1): k in [0..Floor (n/3)]]: n in [0..40]]; // Vincenzo Librandi, Sep 07 2025

Crossrefs

Cf. A002315, A387694, A387696.

Cf. A387692.

Sequence in context: A200528 A127982 A109164 * A212689 A027984 A342313

Adjacent sequences: A387692 A387693 A387694 * A387696 A387697 A387698

Keyword

nonn,easy

Author

Seiichi Manyama, Sep 06 2025

Status

approved