A390826 a(n) = Sum_{k=0..n} binomial(n+2*k+1,n-k) * Fibonacci(k+1).

1, 3, 10, 38, 150, 592, 2325, 9110, 35679, 139748, 547425, 2144474, 8400768, 32909089, 128917478, 505018715, 1978350057, 7749948948, 30359496230, 118929689144, 465892807802, 1825079250915, 7149529278034, 28007424262257, 109715728594068, 429798220237917

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

G.f.: (1 - x)^4 / (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+1, 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+1, n-k)*fibonacci(k+1));
(Magma) [&+[Binomial(n+2*k+1, n-k)*Fibonacci(k+1): k in [0..n]] : n in [0..40] ]; // Vincenzo Librandi, Nov 27 2025

Crossrefs

Partial sums of A390825.

Cf. A000045, A000217.

Sequence in context: A149047 A196469 A083692 * A308124 A151059 A151060

Adjacent sequences: A390823 A390824 A390825 * A390827 A390828 A390829

Keyword

nonn,easy

Author

Seiichi Manyama, Nov 20 2025

Status

approved