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

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, 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, -258, 0, 172, 0, 170, 0, -516, 0, 454, 0, 187, 0, -989, 0, 1095, 0, 40, 0, -1811, 0, 2487

Offset: 0

Author

Joerg Arndt, Oct 18 2012

Keywords

• sign

Links

• Robert Israel, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A101912, A238429.

Programs

Maple

P:= 1+x: d:= 1:
while d < 127 do
  P:= convert(series(1+x/subs(x=x^2,P),x,2+2*d),polynom);
  d:= 1+2*d;
od:
seq(coeff(P,x,i),i=0..d); # Robert Israel, Mar 13 2018

Mathematica

nmax = 75; sol = {a[0] -> 1};
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}];
sol /. Rule -> Set;
a /@ Range[0, nmax] (* Jean-François Alcover, Nov 01 2019 *)

PARI

N=166;  R=O('x^N);  x='x+R;
A= 1; for (k=1,N+1, A = 1 + x / subst(A,'x,'x^2) + R; );
Vec(A)

Formula

G.f. A(x) = 1/B(x) = 1 + x*B(x^2) where B(x) is the g.f. of A101912.

G.f.: 1+x/(1+x^2/(1+x^4/(1+x^8/(1+ ...)))) (continued fraction).

(A(x) + 1) / (A(x) - 1) = 1 + 2*A(x^2) / x. [Joerg Arndt, Feb 28 2014]

A(x^3) = F(x) - x where F(x) is the g.f. of A238429. [Joerg Arndt, Feb 28 2014]