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G.f.: A(x) = 1/(1 - x*B(x^2)), where B(x) = Sum_{n>=0} a(n)^2*x^n is the g.f. of A121648.
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%I #7 May 18 2014 02:36:06

%S 1,1,1,2,3,5,8,16,27,51,89,170,300,564,1008,1972,3563,6847,12483,

%T 24340,44583,86071,158600,309554,572548,1108068,2057792,4003278,

%U 7451924,14415482,26913176,52545636,98321435,190858943,358017691,698449146

%N G.f.: A(x) = 1/(1 - x*B(x^2)), where B(x) = Sum_{n>=0} a(n)^2*x^n is the g.f. of A121648.

%F a(n) = A121648(n)^(1/2).

%F G.f. satisfies: A(x)*A(-x) = ( A(x) + A(-x) )/2. - _Paul D. Hanna_, Aug 14 2006

%e A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 8*x^6 + 16*x^7 + 27*x^8 +...

%e The coefficients of 1 - 1/A(x) equal the square of each term:

%e 1/A(x) = 1 - x - x^3 - x^5 - 4*x^7 - 9*x^9 - 25*x^11 - 64*x^13 - 256*x^15 -... - a(n)^2*x^(2*n+1) -...

%o (PARI) {a(n)=local(B);if(n==0,1,B=sum(k=0,n\2,a(k)^2*x^(2*k)); polcoeff(1/(1-x*B+x*O(x^n)),n))}

%o (PARI) {a(n)=local(A,m); if(n<0, 0, m=1; A=1+x+O(x^2); while(m<=n, m*=2; A=1/(1-x*sum(k=0,m-1, polcoeff(A,k)^2*x^(2*k), O(x^(2*m))))); polcoeff(A,n))} /* _Michael Somos_, Aug 18 2006 */

%Y Cf. A121648; bisections: A121650, A121651.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Aug 14 2006