A390857 a(n) = Sum_{k=0..n} (-1)^k * binomial(n+2*k+2,n-k) * Fibonacci(k+1).

1, 2, 2, 4, 18, 58, 143, 337, 868, 2356, 6289, 16367, 42435, 110938, 291366, 764147, 1999478, 5230106, 13690840, 35853360, 93880186, 245769487, 643381907, 1684376234, 4409868188, 11545349723, 30225984563, 79132089737, 207170284648, 542380110476, 1419971393049

Offset

0,2

Links

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

Index entries for linear recurrences with constant coefficients, signature (5,-11,17,-14,6,-1).

Formula

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

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

a(n) = 5*a(n-1) - 11*a(n-2) + 17*a(n-3) - 14*a(n-4) + 6*a(n-5) - a(n-6).

Mathematica

Table[Sum[(-1)^k*Binomial[n+2*k+2, 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+2*k+2, n-k)*fibonacci(k+1));
(Magma) [&+[(-1)^k*Binomial(n+2*k+2, n-k)*Fibonacci(k+1): k in [0..n]] : n in [0..40] ]; // Vincenzo Librandi, Nov 26 2025

Crossrefs

Partial sums of A390856.

Cf. A000045, A000217, A390827.

Sequence in context: A232161 A335684 A052628 * A006853 A120417 A175185

Adjacent sequences: A390854 A390855 A390856 * A390858 A390859 A390860

Keyword

nonn,easy

Author

Seiichi Manyama, Nov 21 2025

Status

approved