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G.f. satisfies A(x) = f(x) + x*A(x)*f(x)^3, where f(x) = Sum_{k>=0} x^((4^k-1)/3).
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%I #12 Oct 11 2020 05:52:01

%S 1,2,5,14,36,96,254,676,1792,4756,12621,33490,88868,235818,625764,

%T 1660510,4406296,11692452,31026836,82332140,218474784,579739960,

%U 1538385398,4082226194,10832507040,28744906148,76276860598,202406625820

%N G.f. satisfies A(x) = f(x) + x*A(x)*f(x)^3, where f(x) = Sum_{k>=0} x^((4^k-1)/3).

%F a(n) = A087221(3n+1).

%e Given f(x) = 1 + x + x^5 + x^21 + x^85 + x^341 + ...

%e so that f(x)^3 = 1 + 3x + 3x^2 + x^3 + 3x^5 + 6x^6 + 3x^7 + 3x^10 + ...

%e then A(x) = (1 + x + x^5 + ...) + x*A(x)*(1 + 3x + 3x^2 + x^3 + 3x^5 + 6x^6 + ...)

%e = 1 + 2x + 5x^2 + 14x^3 + 36x^4 + 96x^5 + 254x^6 + ...

%o (PARI) a(n)=local(A,m); if(n<1,n==0,m=1; A=1+O(x); while(m<=3*n+3,m*=4; A=1/(1/subst(A,x,x^4)-x)); polcoeff(A,3*n+1))

%Y Cf. A087221, A087222, A087224.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Aug 27 2003