A389933 a(n) = Sum_{k=0..n} binomial(n+k+1,n-k) * Fibonacci(k+1).

1, 3, 9, 29, 96, 319, 1059, 3513, 11651, 38640, 128149, 425007, 1409541, 4674761, 15503904, 51418891, 170531391, 565569477, 1875718199, 6220842720, 20631502201, 68424633483, 226931147409, 752619970949, 2496073488096, 8278258747399, 27454948027659

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,-7,5,-1).

Formula

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

G.f.: (1 - x)^2 / (1 - 5*x + 7*x^2 - 5*x^3 + x^4).

a(n) = 5*a(n-1) - 7*a(n-2) + 5*a(n-3) - a(n-4).

Mathematica

Table[Sum[Binomial[n+k+1, n-k]*Fibonacci[k+1], {k, 0, n}], {n, 0, 40}] (* Vincenzo Librandi, Nov 28 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(n+k+1, n-k)*fibonacci(k+1));
(Magma) [&+[Binomial(n+k+1, n-k)*Fibonacci(k+1): k in [0..n]] : n in [0..40] ]; // Vincenzo Librandi, Nov 28 2025

Crossrefs

Partial sums of A389699.

Cf. A000045, A104630.

Sequence in context: A290897 A289448 A071732 * A289804 A071736 A286955

Adjacent sequences: A389930 A389931 A389932 * A389934 A389935 A389936

Keyword

nonn,easy

Author

Seiichi Manyama, Nov 20 2025

Status

approved