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A218031 G.f. A(x) satisfies A(x) = 1 + x / A(x^2).

%I A218031 #23 Nov 01 2019 11:53:10
%S A218031 1,1,0,-1,0,1,0,0,0,-1,0,1,0,0,0,-2,0,3,0,-1,0,-3,0,6,0,-4,0,-4,0,12,
%T A218031 0,-10,0,-5,0,23,0,-25,0,-2,0,43,0,-57,0,12,0,74,0,-124,0,56,0,120,0,
%U A218031 -258,0,172,0,170,0,-516,0,454,0,187,0,-989,0,1095,0,40,0,-1811,0,2487
%N A218031 G.f. A(x) satisfies A(x) = 1 + x / A(x^2).
%H A218031 Robert Israel, <a href="/A218031/b218031.txt">Table of n, a(n) for n = 0..10000</a>
%F A218031 G.f. A(x) = 1/B(x) = 1 + x*B(x^2) where B(x) is the g.f. of A101912.
%F A218031 G.f.: 1+x/(1+x^2/(1+x^4/(1+x^8/(1+ ...)))) (continued fraction).
%F A218031 (A(x) + 1) / (A(x) - 1) =  1 + 2*A(x^2) / x. [_Joerg Arndt_, Feb 28 2014]
%F A218031 A(x^3) = F(x) - x where F(x) is the g.f. of A238429. [_Joerg Arndt_, Feb 28 2014]
%p A218031 P:= 1+x: d:= 1:
%p A218031 while d < 127 do
%p A218031   P:= convert(series(1+x/subs(x=x^2,P),x,2+2*d),polynom);
%p A218031   d:= 1+2*d;
%p A218031 od:
%p A218031 seq(coeff(P,x,i),i=0..d); # _Robert Israel_, Mar 13 2018
%t A218031 nmax = 75; sol = {a[0] -> 1};
%t A218031 Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x] - (1 + x/A[x^2]) + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
%t A218031 sol /. Rule -> Set;
%t A218031 a /@ Range[0, nmax] (* _Jean-François Alcover_, Nov 01 2019 *)
%o A218031 (PARI)
%o A218031 N=166;  R=O('x^N);  x='x+R;
%o A218031 A= 1; for (k=1,N+1, A = 1 + x / subst(A,'x,'x^2) + R; );
%o A218031 Vec(A)
%Y A218031 Cf. A101912, A238429.
%K A218031 sign
%O A218031 0,16
%A A218031 _Joerg Arndt_, Oct 18 2012