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 A346506 #23 Jan 08 2023 10:26:27
%S A346506 1,2,5,17,66,274,1190,5341,24577,115326,549747,2654739,12959468,
%T A346506 63848307,317064921,1585380283,7975134892,40332823042,204947059412,
%U A346506 1045859173864,5357606584326,27540884494209,142023060613755,734506610474205,3808771672620618,19798640525731461,103149287155802941
%N A346506 G.f. A(x) satisfies: A(x) = (1 + x * A(x)^2) / (1 - x + x^2).
%F A346506 a(0) = 1, a(1) = 2; a(n) = 2 * a(n-1) + a(n-2) + Sum_{k=2..n-1} a(k) * a(n-k-1).
%F A346506 From _Nikolaos Pantelidis_, Jan 08 2023 (Start)
%F A346506 G.f.: 1/G(0), where G(k) = 1-(2*x-x^2)/(1-x/G(k+1)) (continued fraction).
%F A346506 G.f.: (1-x+x^2-sqrt(x^4-2*x^3+3*x^2-6*x+1))/(2*x).
%F A346506 (End)
%t A346506 nmax = 26; A[_] = 0; Do[A[x_] = (1 + x A[x]^2)/(1 - x + x^2) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t A346506 a[0] = 1; a[1] = 2; a[n_] := a[n] = 2 a[n - 1] + a[n - 2] + Sum[a[k] a[n - k - 1], {k, 2, n - 1}]; Table[a[n], {n, 0, 26}]
%Y A346506 Cf. A004148, A006318, A025244, A025258, A078482, A171199, A175934, A307733, A346505.
%K A346506 nonn
%O A346506 0,2
%A A346506 _Ilya Gutkovskiy_, Jul 21 2021