A390852 a(n) = Sum_{k=0..n} (-1)^k * binomial(n+k+1,n-k) * Fibonacci(k+1).

1, 1, 1, 3, 8, 17, 35, 75, 163, 352, 757, 1629, 3509, 7559, 16280, 35061, 75511, 162631, 350263, 754368, 1624697, 3499145, 7536185, 16230843, 34956712, 75287017, 162147259, 349220019, 752122619, 1619862560, 3488732621, 7513758021, 16182541261, 34852685023

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

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

Crossrefs

Partial sums of A390851.

Cf. A000045, A389933.

Sequence in context: A159217 A052996 A112523 * A369734 A147419 A106691

Adjacent sequences: A390849 A390850 A390851 * A390853 A390854 A390855

Keyword

nonn,easy

Author

Seiichi Manyama, Nov 21 2025

Status

approved