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%I #11 Aug 28 2023 10:52:16
%S 1,1,1,1,2,5,11,22,45,99,226,515,1168,2670,6186,14467,33985,80105,
%T 189636,451060,1077225,2580979,6201602,14942480,36098349,87417956,
%U 212159347,515937882,1257048536,3068146679,7500995555,18366760161,45037590888,110588510089
%N G.f. satisfies A(x) = 1 + x*A(x)/(1 - x^3*A(x)^2).
%F a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k-1,k) * binomial(n-k+1,n-3*k)/(n-k+1).
%o (PARI) a(n) = sum(k=0, n\3, binomial(n-2*k-1, k)*binomial(n-k+1, n-3*k)/(n-k+1));
%Y Cf. A101785, A106228, A112805.
%K nonn
%O 0,5
%A _Seiichi Manyama_, Aug 28 2023