A101912 does not have a rational g.f.

[Ralf Stephan]

You wrote 
> A comment by Ralf Stephan to A101912 says "Sequence appears to have a 
> rational g.f.".  Actually this g.f., which by definition satisfies
> A(x) = 1/(1+x*A(x^2)), can't be  rational: if A(x) was rational, we'd have 
> A(x) ~ c x^n as x -> infinity for
> some nonzero constant c and some integer n.  But if n >= 0,
> 1/(1+x*A(x^2)) ~ 1/(c x^(2n+1)) as x -> infinity, while if n <= -1, 
> 1/(1+x*A(x^2)) ~ 1.  Both cases are incompatible with A(x) ~ c x^n.
> I think the confusion is due to the fact that
> (1 + x^2 + x^4 + x^8 + x^10)/
> (1 + x + x^2 + x^4 + x^5 + x^8 + x^9 + x^10) = A(x) + O(x^31).
> But the coefficients of x^31 are different: 454 for A(x) versus
> 455 for that rational function.

Thanks. So we would have a refutation of the conjecture.
At the same time, that rational g.f. is a good approximation
to A(x), and the sequence of it should be entered into the
OEIS, too.

Regards,
ralf