This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A346503 #14 Sep 26 2024 03:27:11
%S A346503 1,0,0,1,1,1,3,5,7,14,26,43,79,148,264,483,903,1664,3080,5771,10795,
%T A346503 20209,38059,71799,135569,256762,487310,925981,1762841,3361897,
%U A346503 6419595,12275301,23505143,45061424,86485016,166176499,319630115,615387675,1185940209,2287527119,4416083429
%N A346503 G.f. A(x) satisfies A(x) = 1 + x^3 * A(x)^2 / (1 - x).
%F A346503 a(0) = 1, a(1) = a(2) = 0; a(n) = a(n-1) + Sum_{k=0..n-3} a(k) * a(n-k-3).
%F A346503 a(n) ~ 2^(n+1) / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Jul 30 2021
%F A346503 From _Seiichi Manyama_, Sep 26 2024: (Start)
%F A346503 G.f.: 2/(1 + sqrt(1 - 4*x^3/(1 - x))).
%F A346503 a(n) = Sum_{k=0..floor(n/3)} binomial(2*k,k) * binomial(n-2*k-1,n-3*k) / (k+1). (End)
%t A346503 nmax = 40; A[_] = 0; Do[A[x_] = 1 + x^3 A[x]^2/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t A346503 a[0] = 1; a[1] = a[2] = 0; a[n_] := a[n] = a[n - 1] + Sum[a[k] a[n - k - 3], {k, 0, n - 3}]; Table[a[n], {n, 0, 40}]
%Y A346503 Cf. A002212, A023426, A023431, A090345, A216604 (partial sums), A346504, A366588, A376490.
%K A346503 nonn
%O A346503 0,7
%A A346503 _Ilya Gutkovskiy_, Jul 21 2021