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A307970 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3*A(x)^2.

%I A307970 #12 Jul 03 2021 03:21:22
%S A307970 1,1,1,1,2,3,4,7,12,19,32,56,96,165,290,512,902,1601,2862,5124,9198,
%T A307970 16585,29990,54336,98702,179742,327942,599432,1097756,2013737,3699596,
%U A307970 6806866,12541518,23137270,42736850,79031394,146309968,271142469,502978944,933921458,1735634266
%N A307970 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3*A(x)^2.
%C A307970 Shifts 3 places left when convolved with itself.
%H A307970 Seiichi Manyama, <a href="/A307970/b307970.txt">Table of n, a(n) for n = 0..3000</a>
%F A307970 G.f.: 1/(1 - x/(1 - x^3/(1 - x^3/(1 - x/(1 - x^3/(1 - x^3/(1 - x/(1 - x^3/(1 - x^3/(1 - ...)))))))))), a continued fraction.
%F A307970 Recurrence: a(n+3) = Sum_{k=0..n} a(k)*a(n-k).
%F A307970 a(n) ~ sqrt(3 + 4*r^4 + 8*r^5) / (4*sqrt(Pi)*n^(3/2)*r^(n+3)), where r = 0.51899425841331458784223152875297289010563957455264491744143... is the root of the equation 1 + r + r^2 = 1/(4*r^3). - _Vaclav Kotesovec_, Jul 03 2021
%e A307970 G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 7*x^7 + 12*x^8 + 19*x^9 + 32*x^10 + ...
%p A307970 a:= proc(n) option remember; `if`(n<3, 1,
%p A307970       add(a(j)*a(n-3-j), j=0..n-3))
%p A307970     end:
%p A307970 seq(a(n), n=0..50);  # _Alois P. Heinz_, May 08 2019
%t A307970 terms = 40; A[_] = 0; Do[A[x_] = 1 + x + x^2 + x^3 A[x]^2 + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]
%t A307970 a[n_] := a[n] = Sum[a[k] a[n - k - 3], {k, 0, n - 3}]; a[0] = a[1] = a[2] = 1; Table[a[n], {n, 0, 40}]
%Y A307970 Cf. A000108, A007477, A307971, A307972.
%K A307970 nonn
%O A307970 0,5
%A A307970 _Ilya Gutkovskiy_, May 08 2019