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A358547 a(n) = Sum_{k=0..floor(n/3)} (n-k)!/(n-3*k)!.

%I A358547 #21 Nov 25 2022 08:01:23
%S A358547 1,1,1,3,7,13,45,151,403,1617,6793,23275,105951,522133,2159077,
%T A358547 10964223,61134955,293587801,1641566913,10124731987,55014334903,
%U A358547 335177088285,2251814156701,13587321392743,89436553249347,647267633012833,4276528756374265,30198747030078651
%N A358547 a(n) = Sum_{k=0..floor(n/3)} (n-k)!/(n-3*k)!.
%H A358547 Seiichi Manyama, <a href="/A358547/b358547.txt">Table of n, a(n) for n = 0..673</a>
%F A358547 a(n) = (3 * (2*n-1) * a(n-1) - n * a(n-2) + 2 * (n-1) * n * (2*n-3) * a(n-3) + 2 * (2*n-3))/(9 * (n-1)) for n > 2.
%F A358547 a(n) ~ sqrt(Pi) * 2^(2*n/3 + 1) * n^(2*n/3 + 1/2) / (3^(2*n/3 + 3/2) * exp(2*n/3 - (2/3)^(1/3) * n^(1/3))) * (1 + 1/(2^(4/3) * 3^(5/3) * n^(1/3)) + 145/(2^(11/3) * 3^(10/3) * n^(2/3)) + 3349/(23328*n)). - _Vaclav Kotesovec_, Nov 25 2022
%o A358547 (PARI) a(n) = sum(k=0, n\3, (n-k)!/(n-3*k)!);
%Y A358547 Cf. A122852, A330044, A357532.
%K A358547 nonn
%O A358547 0,4
%A A358547 _Seiichi Manyama_, Nov 21 2022