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 A358560 #18 Nov 25 2022 09:46:10
%S A358560 1,1,1,3,7,13,33,91,223,597,1753,4963,14391,44413,137137,427083,
%T A358560 1382383,4534981,14981673,50719507,174494983,605276973,2135204161,
%U A358560 7647369403,27643067007,101211363253,375548195833,1406858084931,5326762882903,20403498329437
%N A358560 a(n) = Sum_{k=0..floor(n/3)} (n-k)!/(k! * (n-3*k)!).
%H A358560 Seiichi Manyama, <a href="/A358560/b358560.txt">Table of n, a(n) for n = 0..1000</a>
%F A358560 a(n) = (4 * a(n-1) - a(n-2) + 2 * (2*n-3) * a(n-3))/3 for n > 2.
%F A358560 a(n) ~ c * 2^(2*n/3) * n^(n/3) / (3^(n/3) * exp(n/3 - 2^(1/3) * n^(2/3) / 3^(2/3) + n^(1/3) / (2^(4/3) * 3^(7/3)))) * (1 + 7795/(5832*6^(2/3)*n^(1/3)) + 135724109/(2040733440*6^(1/3)*n^(2/3)) - 5962064767253/(42845606719488*n)), where c = 0.46562048925..., conjecture: c = sqrt(2) * exp(-1/81) / 3. - _Vaclav Kotesovec_, Nov 25 2022
%o A358560 (PARI) a(n) = sum(k=0, n\3, (n-k)!/(k!*(n-3*k)!));
%Y A358560 Cf. A000930, A118395, A358547.
%K A358560 nonn,easy
%O A358560 0,4
%A A358560 _Seiichi Manyama_, Nov 22 2022