%I #15 Apr 02 2013 20:40:31
%S 1,2,5,4,14,10,28,8,69,28,116,20,252,56,340,16,726,138,916,56,1982,
%T 232,2120,40,4844,504,4860,112,11320,680,9520,32,24525,1452,19508,276,
%U 51636,1832,34636,112,104388,3964,67480,464,203676,4240,110288,80,388732,9688,206908,1008
%N G.f. satisfies: A(x) = (A(x^2) + x)^2.
%H Paul D. Hanna, <a href="/A224272/b224272.txt">Table of n, a(n) for n = 0..1024</a>
%F a(m*2^n-1) = a(m-1)*2^n for n>=0, m>=1:
%F a(2^n-1) = 2^n, a(3*2^n-1) = 5*2^n, a(5*2^n-1) = 14*2^n, for n>=0.
%F a(m) is odd iff m = 2*4^n (n>=0) or m=0.
%F a(2*4^n) == 5 (mod 8) for n>=0.
%e G.f.: A(x) = 1 + 2*x + 5*x^2 + 4*x^3 + 14*x^4 + 10*x^5 + 28*x^6 + 8*x^7 + 69*x^8 + 28*x^9 + 116*x^10 + 20*x^11 + 252*x^12 +...
%e where
%e A(x)^(1/2) = 1 + x + 2*x^2 + 5*x^4 + 4*x^6 + 14*x^8 + 10*x^10 + 28*x^12 + 116*x^20 + 20*x^22 + 252*x^24 +...
%e A(x)^2 = 1 + 4*x + 14*x^2 + 28*x^3 + 69*x^4 + 116*x^5 + 252*x^6 + 340*x^7 +...
%o (PARI) {a(n)=local(A=1+x); for(i=1, #binary(n+1), A=(subst(A, x, x^2) + x +x*O(x^n))^2); polcoeff(A, n, x)}
%o for(n=0, 64, print1(a(n), ", "))
%K nonn
%O 0,2
%A _Paul D. Hanna_, Apr 02 2013