A390853 a(n) = Sum_{k=0..n} (-1)^k * binomial(n+k+2,n-k) * Fibonacci(k+1).

1, 2, 3, 6, 14, 31, 66, 141, 304, 656, 1413, 3042, 6551, 14110, 30390, 65451, 140962, 303593, 653856, 1408224, 3032921, 6532066, 14068251, 30299094, 65255806, 140542823, 302690082, 651910101, 1404032720, 3023895280, 6512627901, 14026385922, 30208927183, 65061612206

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

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

Crossrefs

Partial sums of A390852.

Cf. A000045, A390824, A390859.

Sequence in context: A211931 A264078 A006444 * A335242 A032047 A032065

Adjacent sequences: A390850 A390851 A390852 * A390854 A390855 A390856

Keyword

nonn,easy

Author

Seiichi Manyama, Nov 21 2025

Status

approved