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%I #18 Oct 03 2023 10:01:00
%S 1,1,2,3,6,11,21,39,75,141,273,519,1009,1933,3770,7263,14202,27479,
%T 53846,104543,205216,399543,785460,1532779,3017106,5899167,11624580,
%U 22766607,44905518,88073091,173863965,341425551,674506059,1326019653,2621371005,5158412943,10203609597
%N a(n) = Sum_{i=0..floor(q(n)/3)} binomial(n-3*(i+1), q(n)-3*i) with q(n) = ceiling((n-3)/2).
%H Gábor Czédli, <a href="https://arxiv.org/abs/2309.13783">Minimum-sized generating sets of the direct powers of the free distributive lattice on three generators and a Sperner theorem</a>, arXiv:2309.13783 [math.CO], 2023. See formulas (3.5) at p. 4 and (4.15) at p. 8.
%F From Remark 3.4 at p. 5 in Czédli: (Start)
%F A366108(n)/a(n) ~ 7/4.
%F A366109(n)/a(n) ~ 7/6. (End)
%F a(n) ~ c*2^(n+1)/sqrt(n), with c = 1/(7*sqrt(2*Pi)) = (2/7)* A218708.
%t q[n_]:=Ceiling[(n-3)/2]; a[n_]:=Sum[Binomial[n-3(i+1),q[n]-3i], {i,0,Floor[q[n]/3]}]; Array[a,37,3]
%o (PARI) a(n) = my(q=ceil((n-3)/2)); sum(i=0, q\3, binomial(n-3*(i+1), q-3*i)); \\ _Michel Marcus_, Sep 30 2023
%Y Cf. A004526, A218708, A366108, A366109.
%K nonn
%O 3,3
%A _Stefano Spezia_, Sep 29 2023