A389699 a(n) = Sum_{k=0..n} binomial(n+k,n-k) * Fibonacci(k+1).

1, 2, 6, 20, 67, 223, 740, 2454, 8138, 26989, 89509, 296858, 984534, 3265220, 10829143, 35914987, 119112500, 395038086, 1310148722, 4345124521, 14410659481, 47793131282, 158506513926, 525688823540, 1743453517147, 5782185259303, 19176689280260, 63599728348374

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

G.f.: (1 - x)^3 / (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, 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, n-k)*fibonacci(k+1));
(Magma) [&+[Binomial(n+k, n-k)*Fibonacci(k+1): k in [0..n]] : n in [0..40] ]; // Vincenzo Librandi, Nov 28 2025

Crossrefs

Cf. A000027, A000045, A104630, A289780, A389933.

Sequence in context: A193234 A285197 A372872 * A148475 A148476 A148477

Adjacent sequences: A389696 A389697 A389698 * A389700 A389701 A389702

Keyword

nonn,easy

Author

Seiichi Manyama, Nov 20 2025

Status

approved