A390827 a(n) = Sum_{k=0..n} binomial(n+2*k+2,n-k) * Fibonacci(k+1).

1, 4, 14, 52, 202, 794, 3119, 12229, 47908, 187656, 735081, 2879555, 11280323, 44189412, 173106890, 678125605, 2656475662, 10406424610, 40765920840, 159695609984, 625588417786, 2450667668701, 9600196946735, 37607621208992, 147323349803060, 577121570040977

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

G.f.: (1 - x)^3 / (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+2, 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+2, n-k)*fibonacci(k+1));
(Magma) [&+[Binomial(n+2*k+2, n-k)*Fibonacci(k+1): k in [0..n]] : n in [0..40] ]; // Vincenzo Librandi, Nov 27 2025

Crossrefs

Partial sums of A390826.

Cf. A000045, A000217.

Sequence in context: A199698 A052710 A364410 * A262594 A345242 A370891

Adjacent sequences: A390824 A390825 A390826 * A390828 A390829 A390830

Keyword

nonn,easy

Author

Seiichi Manyama, Nov 20 2025

Status

approved