A390859 a(n) = Sum_{k=0..n} (-1)^k * binomial(n+k+3,n-k) * Fibonacci(k+1).

1, 3, 6, 12, 26, 57, 123, 264, 568, 1224, 2637, 5679, 12230, 26340, 56730, 122181, 263143, 566736, 1220592, 2628816, 5661737, 12193803, 26262054, 56561148, 121816954, 262359777, 565049859, 1216959960, 2620992680, 5644887960, 12157515861, 26183901783, 56392828966

Offset

0,2

Links

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

Index entries for linear recurrences with constant coefficients, signature (3,-3,3,-1).

Formula

G.f.: 1/((1-x)^4 * (1+g-g^2)), where g = x/(1-x)^2.

G.f.: 1 / (1 - 3*x + 3*x^2 - 3*x^3 + x^4).

a(n) = 3*a(n-1) - 3*a(n-2) + 3*a(n-3) - a(n-4).

Mathematica

Table[Sum[(-1)^k*Binomial[n+k+3, n-k]*Fibonacci[k+1], {k, 0, n}], {n, 0, 40}] (* Vincenzo Librandi, Nov 26 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, (-1)^k*binomial(n+k+3, n-k)*fibonacci(k+1));
(Magma) [&+[(-1)^k*Binomial(n+k+3, n-k)*Fibonacci(k+1): k in [0..n]] : n in [0..40] ]; // Vincenzo Librandi, Nov 26 2025

Crossrefs

Partial sums of A390853.

Cf. A000045, A099444.

Sequence in context: A274059 A092886 A207094 * A135035 A054195 A054190

Adjacent sequences: A390856 A390857 A390858 * A390860 A390861 A390862

Keyword

nonn,easy

Author

Seiichi Manyama, Nov 21 2025

Status

approved