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A390850

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

1, -1, 0, 2, 3, 4, 9, 22, 48, 101, 216, 467, 1008, 2170, 4671, 10060, 21669, 46670, 100512, 216473, 466224, 1004119, 2162592, 4657618, 10031211, 21604436, 46529937, 100212518, 215829840, 464837341, 1001130120, 2156155339, 4643757840, 10001360522, 21540143943, 46391468444

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OFFSET

0,4

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

G.f.: (1 - x)^4 / (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) for n > 4.

MATHEMATICA

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