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%I #8 Jul 26 2018 09:10:55
%S 1,1,0,0,1,-1,2,-2,2,0,-4,12,-24,38,-46,33,29,-176,443,-827,1222,
%T -1310,433,2488,-8814,19528,-33599,44928,-37805,-17916,168049,-463252,
%U 921694,-1446018,1679053,-808620,-2598482,10515127,-24690122,44515322,-62719429,58496244,10670109,-213311788,632128236
%N G.f. A(x) defined by: A(x)^3 consists entirely of integer coefficients between 1 and 3 (A083953); A(x) is the unique power series solution with A(0)=1.
%C Does limit_{n ->infinity} a(n)/a(n+1) exist?
%H N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.NT/0509316">On the Integrality of n-th Roots of Generating Functions</a>, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
%t kmax = 45;
%t A[x_] = Sum[a[k] x^k, {k, 0, kmax}];
%t coes = CoefficientList[A[x]^3 + O[x]^kmax , x];
%t r = {};
%t Do[r = Flatten @ Append[r, Reduce[1 <= coes[[k]] <= 3, a[k-1], Integers] // ToRules]; coes = coes /. r, {k, 1, kmax}];
%t Array[a, kmax, 0] /. r (* _Jean-François Alcover_, Jul 26 2018 *)
%Y Cf. A083953, A084202, A084204-A084212.
%K sign
%O 0,7
%A _Paul D. Hanna_, May 19 2003