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 A349186 #5 Nov 11 2021 20:50:21
%S A349186 1,1,3,8,21,57,157,438,1237,3530,10165,29505,86243,253654,750157,
%T A349186 2229469,6655369,19946979,60000443,181076982,548125929,1663786344,
%U A349186 5063133335,15444046031,47211447131,144614092732,443803262627,1364370846941,4201333752921,12957168021207
%N A349186 G.f. A(x) satisfies: A(x) = (1 - x) / (1 - 2 * x - x^2 - x^3 * A(x)).
%F A349186 G.f.: (1 - 2*x - x^2 - sqrt(1 - 4*x + 2*x^2 + 5*x^4)) / (2*x^3).
%F A349186 a(0) = a(1) = 1; a(n) = 2 * a(n-1) + a(n-2) + Sum_{k=0..n-3} a(k) * a(n-k-3).
%t A349186 nmax = 29; A[_] = 0; Do[A[x_] = (1 - x)/(1 - 2 x - x^2 - x^3 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
%t A349186 nmax = 29; CoefficientList[Series[(1 - 2 x - x^2 - Sqrt[1 - 4 x + 2 x^2 + 5 x^4])/(2 x^3), {x, 0, nmax}], x]
%t A349186 a[0] = a[1] = 1; a[n_] := a[n] = 2 a[n - 1] + a[n - 2] + Sum[a[k] a[n - k - 3], {k, 0, n - 3}]; Table[a[n], {n, 0, 29}]
%Y A349186 Cf. A002026, A004148, A086581, A217333, A349185.
%K A349186 nonn
%O A349186 0,3
%A A349186 _Ilya Gutkovskiy_, Nov 09 2021