A357532 a(n) = Sum_{k=0..floor(n/3)} (n-2*k)!/(n-3*k)!.

1, 1, 1, 2, 3, 4, 7, 12, 19, 34, 63, 112, 211, 414, 799, 1588, 3267, 6706, 13999, 30024, 64723, 141142, 314271, 705724, 1599619, 3685338, 8573167, 20112016, 47804499, 114743614, 277615903, 679057092, 1676636611, 4171532674, 10477002159, 26545428568, 67755344467, 174386589606

Offset: 0

Author

Seiichi Manyama, Nov 19 2022

Keywords

• nonn
• easy

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A072374, A122852, A357533, A357570.

Programs

PARI

a(n) = sum(k=0, n\3, (n-2*k)!/(n-3*k)!);

Formula

a(n) = (2 * a(n-1) + n * a(n-3) + 1)/3 for n > 2.

a(n) ~ c * n^(n/3 + 1/2) / (3^(n/3) * exp(n/3 - n^(2/3)/3^(2/3) - 2*n^(1/3) / 3^(7/3))) * (1 + 1235/(729 * 3^(2/3) * n^(1/3)) + 9452027/(15943230 * 3^(1/3) * n^(2/3)) + 16015315669/(41841412812*n)), where c = 0.50682110703119..., conjecture: c = exp(4/81) * sqrt(2*Pi) / 3^(3/2). - Vaclav Kotesovec, Nov 25 2022