A101912 does not have a rational g.f.

[Robert Israel]

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.


Robert Israel                                israel at math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            Vancouver, BC, Canada