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 A349185 #6 Nov 11 2021 20:48:55
%S A349185 1,1,4,11,35,111,365,1221,4160,14371,50251,177503,632514,2271027,
%T A349185 8208259,29840993,109049568,400352639,1475929092,5461571729,
%U A349185 20279092033,75531360153,282123848574,1056539226257,3966214054639,14922195004703,56258116929483,212505815364639,804142811583006
%N A349185 G.f. A(x) satisfies: A(x) = (1 - x) / (1 - 2 * x - x^2 - x^2 * A(x)).
%F A349185 G.f.: (1 - 2*x - x^2 - sqrt(1 - 4*x - 2*x^2 + 8*x^3 + x^4)) / (2*x^2).
%F A349185 a(0) = a(1) = 1; a(n) = 2 * a(n-1) + a(n-2) + Sum_{k=0..n-2} a(k) * a(n-k-2).
%t A349185 nmax = 28; A[_] = 0; Do[A[x_] = (1 - x)/(1 - 2 x - x^2 - x^2 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t A349185 nmax = 28; CoefficientList[Series[(1 - 2 x - x^2 - Sqrt[1 - 4 x - 2 x^2 + 8 x^3 + x^4])/(2 x^2), {x, 0, nmax}], x]
%t A349185 a[0] = a[1] = 1; a[n_] := a[n] = 2 a[n - 1] + a[n - 2] + Sum[a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 28}]
%Y A349185 Cf. A056010, A085139, A086581, A128720, A349186.
%K A349185 nonn
%O A349185 0,3
%A A349185 _Ilya Gutkovskiy_, Nov 09 2021