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%I #10 Apr 01 2021 12:20:53
%S 1,4,2,0,3,0,2,0,1,0,0,0,0,0,2,0,0,0,2,0,0,0,0,0,0,0,2,0,3,0,0,0,3,0,
%T 2,0,3,0,2,0,0,0,0,0,1,0,0,0,1,0,0,0,3,0,0,0,3,0,0,0,3,0,0,0,0,0,2,0,
%U 3,0,2,0,1,0,0,0,3,0,0,0,3,0,0,0,3,0,2,0,1,0,2,0,3,0,0,0,1,0,2,0,0,0,0,0,1
%N Coefficients of g.f. A(x) where 0 <= a(n) <= 3 for all n>1, with initial terms {1,4}, such that A(x)^(1/4) consists entirely of integer coefficients.
%C Equals the self-convolution 4th power of A106221. What is the frequency of occurrence of the nonzero digits?
%H Antti Karttunen, <a href="/A106220/b106220.txt">Table of n, a(n) for n = 0..1200</a>
%F A(z)=0 at z=-0.30239090673234876830066191989552890839853849934485...
%e A(x) = 1 + 4*x + 2*x^2 + 3*x^4 + 2*x^6 + x^8 + 2*x^14 +...
%e A(x)^(1/4) = 1 + x - x^2 + 2*x^3 - 4*x^4 + 10*x^5 - 26*x^6 +-...
%e A106221 = {1,1,-1,2,-4,10,-26,71,-199,569,-1652,...}.
%o (PARI) {a(n)=local(A=1+4*x);if(n==0,1, for(j=1,n, for(k=0,3,t=polcoeff((A+k*x^j+x*O(x^j))^(1/4),j); if(denominator(t)==1,A=A+k*x^j;break))); return(polcoeff(A+x*O(x^n),n)))}
%Y Cf. A106221, A106216, A106222, A106224, A106226.
%K nonn
%O 0,2
%A _Paul D. Hanna_, May 01 2005