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%I #6 Feb 05 2026 09:20:19
%S 1,1,1,1,3,6,10,19,39,78,154,311,639,1315,2713,5639,11793,24750,52118,
%T 110169,233673,497025,1059939,2266089,4855991,10427740,22436304,
%U 48362711,104427395,225845804,489170840,1061015979,2304411891,5011201284,10910310700
%N G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 * A(x)^2 / (1 - x).
%F G.f.: (1 - x - sqrt(1 - x * (2 - x + 4*x^2 - 4*x^5))) / (2*x^3).
%F a(0) = a(1) = a(2) = 1; a(n) = Sum_{j=0..n-3} Sum_{i=0..j} a(i) * a(j-i).
%t nmax = 34; A[_] = 0; Do[A[x_] = 1 + x + x^2 + x^3 A[x]^2/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t nmax = 34; CoefficientList[Series[(1 - x - Sqrt[1 - x (2 - x + 4 x^2 - 4 x^5)])/(2 x^3), {x, 0, nmax}], x]
%t a[0] = a[1] = a[2] = 1; a[n_] := a[n] = Sum[Sum[a[i] a[j - i], {i, 0, j}], {j, 0, n - 3}]; Table[a[n], {n, 0, 34}]
%Y Cf. A002212, A023431, A025243, A307970, A346503, A346504, A393182.
%K nonn
%O 0,5
%A _Ilya Gutkovskiy_, Feb 04 2026