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a(n) = Sum_{k=0..floor(n/4)} (n-3*k)!/(n-4*k)!.
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%I #26 Nov 25 2022 06:31:58

%S 1,1,1,1,2,3,4,5,8,13,20,29,46,77,128,205,338,581,1012,1733,2990,5293,

%T 9536,17117,30778,56165,104108,193621,360662,677693,1289080,2467373,

%U 4735826,9142837,17814308,34950245,68835118,136197581,271384112,544302973,1096578410,2218459013,4513377436

%N a(n) = Sum_{k=0..floor(n/4)} (n-3*k)!/(n-4*k)!.

%H Seiichi Manyama, <a href="/A357533/b357533.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = (3 * a(n-1) + n * a(n-4) + 1)/4 for n > 3.

%o (PARI) a(n) = sum(k=0, n\4, (n-3*k)!/(n-4*k)!);

%Y Cf. A072374, A122852, A357532, A357570.

%K nonn,easy

%O 0,5

%A _Seiichi Manyama_, Nov 19 2022