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A349365 G.f. A(x) satisfies A(x) = 1 / (1 - x - x * A(x^2)).

%I A349365 #13 Nov 26 2023 08:37:56
%S A349365 1,2,4,10,24,60,148,370,920,2296,5720,14268,35568,88700,221156,551482,
%T A349365 1375096,3428888,8549944,21319624,53160896,132558360,330537528,
%U A349365 824204780,2055176304,5124638944,12778424976,31863351980,79452130896,198116051644,494007751668
%N A349365 G.f. A(x) satisfies A(x) = 1 / (1 - x - x * A(x^2)).
%H A349365 Seiichi Manyama, <a href="/A349365/b349365.txt">Table of n, a(n) for n = 0..1000</a>
%F A349365 G.f.: 1 / (1 - x - x / (1 - x^2 - x^2 / (1 - x^4 - x^4 / (1 - x^8 - x^8 / (1 - ...))))).
%F A349365 a(0) = 1, a(1) = 2; a(n) = a(n-2) + Sum_{k=0..n-1} a(floor(k/2)) * a(n-k-1).
%F A349365 a(n) ~ c * d^n, where d = 2.4935271724548067876965033643037290636931200352851874903211458249308... and c = 0.6156170089558875346518987360369130661426312977478830077668203229773... - _Vaclav Kotesovec_, Nov 16 2021
%F A349365 a(0) = 1; a(n) = a(n-1) + Sum_{k=0..floor((n-1)/2)} a(k) * a(n-1-2*k). - _Seiichi Manyama_, Nov 26 2023
%t A349365 nmax = 30; A[_] = 0; Do[A[x_] = 1/(1 - x - x A[x^2]) + O[x]^(nmax + 1) // Normal,nmax + 1]; CoefficientList[A[x], x]
%t A349365 a[0] = 1; a[1] = 2; a[n_] := a[n] = a[n - 2] + Sum[a[Floor[k/2]] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 30}]
%o A349365 (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=v[i]+sum(j=0, (i-1)\2, v[j+1]*v[i-2*j])); v; \\ _Seiichi Manyama_, Nov 26 2023
%Y A349365 Cf. A000621, A127680.
%Y A349365 Cf. A218032, A319436.
%K A349365 nonn
%O A349365 0,2
%A A349365 _Ilya Gutkovskiy_, Nov 15 2021