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a(0) = 1, a(1) = 0, a(2) = 1; a(n) = a(n-1) + Sum_{k=0..n-3} a(k) * a(n-k-3).
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%I #6 Sep 24 2022 15:00:00

%S 1,0,1,2,2,4,8,13,25,49,91,177,349,681,1349,2693,5377,10806,21820,

%T 44163,89721,182868,373616,765341,1571551,3233690,6667242,13772469,

%U 28498419,59065838,122606998,254865837,530507839,1105663034,2307131590,4819623077,10079039819,21099213611,44211213545

%N a(0) = 1, a(1) = 0, a(2) = 1; a(n) = a(n-1) + Sum_{k=0..n-3} a(k) * a(n-k-3).

%F G.f. A(x) satisfies: A(x) = 1 + x^2 * (1 + x * A(x)^2) / (1 - x).

%t a[0] = 1; a[1] = 0; a[2] = 1; a[n_] := a[n] = a[n - 1] + Sum[a[k] a[n - k - 3], {k, 0, n - 3}]; Table[a[n], {n, 0, 38}]

%t nmax = 38; A[_] = 0; Do[A[x_] = 1 + x^2 (1 + x A[x]^2)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

%Y Cf. A000245, A023426, A023431, A090345, A346503, A346504, A357308.

%K nonn

%O 0,4

%A _Ilya Gutkovskiy_, Sep 23 2022