This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A357532 #30 Nov 25 2022 06:33:49
%S A357532 1,1,1,2,3,4,7,12,19,34,63,112,211,414,799,1588,3267,6706,13999,30024,
%T A357532 64723,141142,314271,705724,1599619,3685338,8573167,20112016,47804499,
%U A357532 114743614,277615903,679057092,1676636611,4171532674,10477002159,26545428568,67755344467,174386589606
%N A357532 a(n) = Sum_{k=0..floor(n/3)} (n-2*k)!/(n-3*k)!.
%H A357532 Seiichi Manyama, <a href="/A357532/b357532.txt">Table of n, a(n) for n = 0..1000</a>
%F A357532 a(n) = (2 * a(n-1) + n * a(n-3) + 1)/3 for n > 2.
%F A357532 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
%o A357532 (PARI) a(n) = sum(k=0, n\3, (n-2*k)!/(n-3*k)!);
%Y A357532 Cf. A072374, A122852, A357533, A357570.
%K A357532 nonn,easy
%O A357532 0,4
%A A357532 _Seiichi Manyama_, Nov 19 2022