A390825 a(n) = Sum_{k=0..n} binomial(n+2*k,n-k) * Fibonacci(k+1).

1, 2, 7, 28, 112, 442, 1733, 6785, 26569, 104069, 407677, 1597049, 6256294, 24508321, 96008389, 376101237, 1473331342, 5771598891, 22609547282, 88570192914, 346963118658, 1359186443113, 5324450027119, 20857894984223, 81708304331811, 320082491643849, 1253884807609401

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (7,-17,23,-16,6,-1).

Formula

G.f.: 1/((1-x) * (1-g-g^2)), where g = x/(1-x)^3.

G.f.: (1 - x)^5 / (1 - 7*x + 17*x^2 - 23*x^3 + 16*x^4 - 6*x^5 + x^6).

a(n) = 7*a(n-1) - 17*a(n-2) + 23*a(n-3) - 16*a(n-4) + 6*a(n-5) - a(n-6).

Mathematica

Table[Sum[Binomial[n+2*k, n-k]*Fibonacci[k+1], {k, 0, n}], {n, 0, 40}] (* Vincenzo Librandi, Nov 27 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(n+2*k, n-k)*fibonacci(k+1));
(Magma) [&+[Binomial(n+2*k, n-k)*Fibonacci(k+1): k in [0..n]] : n in [0..40] ]; // Vincenzo Librandi, Nov 27 2025

Crossrefs

Cf. A000045, A000217, A289797.

Sequence in context: A289607 A068944 A215143 * A289158 A012855 A224066

Adjacent sequences: A390822 A390823 A390824 * A390826 A390827 A390828

Keyword

nonn,easy

Author

Seiichi Manyama, Nov 20 2025

Status

approved