A238429 Expansion of F(x) where F(x) = 1 + x / (1 - x^2 / F(x^2) ).

1, 1, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, -10, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 23, 0, 0, 0, 0, 0, -25, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 43, 0, 0, 0, 0, 0, -57, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 74, 0, 0, 0, 0, 0, -124, 0, 0, 0, 0, 0, 56, 0, 0, 0, 0, 0, 120, 0, 0, 0, 0, 0, -258

Offset: 0

Author

Joerg Arndt, Feb 28 2014

Keywords

• sign

Comments

The sequence of nonzero terms is (apart from the term a(1)=1 here) the same as that for A218031, see the formula "F(x) - x = A(x^3)".

Programs

PARI

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

Formula

G.f.: F(x) = 1+x/ (1-x^2/ (1+x^2/ (1-x^4/ (1+x^4 /(1-x^8/ (1+x^8/ (1-x^16/ ... ))))))) (continued fraction).

F(x) - x = A(x^3) where A(x) is the g.f. of A218031. Note that for G(x) = F(x) - x we have G(x) = 1 + x^3/G(x^2) = 1 + x^3/(1 + x^6 / G(x^2) ) = ... = 1 + x^3/(1 + x^6 / (1 + x^12 / (1 + x^24 / (...) ) ) ) = A(x^3).