A390855 a(n) = Sum_{k=0..n} (-1)^k * binomial(n+2*k,n-k) * Fibonacci(k+1).

1, 0, -1, 2, 12, 26, 45, 109, 337, 957, 2445, 6145, 15990, 42435, 111925, 292353, 762550, 1995297, 5230106, 13701786, 35864306, 93862475, 245723119, 643381907, 1684497627, 4409989581, 11545153305, 30225470334, 79132089737, 207171630917, 542381456745

Offset

0,4

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

G.f.: (1 - x)^5 / ((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, 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, n-k)*fibonacci(k+1));
(Magma) [&+[(-1)^k*Binomial(n+2*k, n-k)*Fibonacci(k+1): k in [0..n]] : n in [0..40] ]; // Vincenzo Librandi, Nov 26 2025

Crossrefs

Partial sums of A390854.

Cf. A000045, A000217, A390825.

Sequence in context: A294554 A098707 A152811 * A294552 A294170 A102960

Adjacent sequences: A390852 A390853 A390854 * A390856 A390857 A390858

Keyword

sign,easy

Author

Seiichi Manyama, Nov 21 2025

Status

approved