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A106216 Coefficients of g.f. A(x) where 0 <= a(n) <= 2 for all n>1, with initial terms {1,3}, such that A(x)^(1/3) consists entirely of integer coefficients.

%I A106216 #4 Mar 30 2012 18:36:45
%S A106216 1,3,0,1,0,0,2,0,0,2,0,0,2,0,0,0,0,0,0,0,0,2,0,0,1,0,0,2,0,0,1,0,0,2,
%T A106216 0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,
%U A106216 0,2,0,0,1,0,0,2,0,0,0,0,0,1,0,0,2,0,0,2,0,0,0,0,0,2,0,0,2,0,0,1,0,0,0,0,0
%N A106216 Coefficients of g.f. A(x) where 0 <= a(n) <= 2 for all n>1, with initial terms {1,3}, such that A(x)^(1/3) consists entirely of integer coefficients.
%C A106216 The self-convolution cube-root equals A106219. Positions of 1's is given by A106217. Positions of 2's is given by A106218. What is the frequency of occurrence of the 1's and 2's?
%F A106216 A(z)=0 at z=-0.322846893915891638743032676733152456643928599...
%e A106216 A(x)^(1/3) = 1 + 1x - 1x^2 + 2x^3 - 4x^4 + 9x^5 - 21x^6 + 53x^6 -+...
%o A106216 (PARI) {a(n)=local(A=1+3*x);if(n==0,1,if(n==1,3, for(j=2,n, for(k=0,2,t=polcoeff((A+k*x^j+x*O(x^j))^(1/3),j); if(denominator(t)==1,A=A+k*x^j;break)));polcoeff(A,n)))}
%Y A106216 Cf. A106217, A106218, A106219, A083953.
%K A106216 nonn
%O A106216 0,2
%A A106216 _Paul D. Hanna_, May 01 2005