A346504 - OEIS

%I #7 Sep 27 2023 05:08:13

%S 1,1,0,1,3,4,6,14,28,49,95,196,386,754,1524,3102,6258,12700,26032,

%T 53440,109772,226457,468863,972300,2020274,4208530,8784556,18365322,

%U 38461110,80682740,169501696,356579216,751138916,1584281062,3345404514,7072055268,14965933024,31702754496

%N G.f. A(x) satisfies: A(x) = 1 + x + x^3 * A(x)^2 / (1 - x).

%F a(0) = a(1) = 1, a(2) = 0; a(n) = a(n-1) + Sum_{k=0..n-3} a(k) * a(n-k-3).

%F G.f.: (1-x)*(1 - sqrt((1 - x - 4*x^3 - 4*x^4)/(1-x))) / (2*x^3). - _Vaclav Kotesovec_, Sep 27 2023

%t nmax = 37; A[_] = 0; Do[A[x_] = 1 + x + x^3 A[x]^2/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

%t a[0] = a[1] = 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, 37}]

%t CoefficientList[Series[(1 - x)*(1 - Sqrt[(1 - x - 4*x^3 - 4*x^4)/(1 - x)]) / (2*x^3), {x, 0, 40}], x] (* _Vaclav Kotesovec_, Sep 27 2023 *)

%Y Cf. A023426, A023431, A025243, A090345, A248100, A257290, A346503.

%K nonn

%O 0,5

%A _Ilya Gutkovskiy_, Jul 21 2021