Let $0 < r < 1$. Consider the sequence $S$ of fractions
$\frac{1}{1+2r+r^2}, \frac{2+2r+2r^2}{2(1+2r+2r^2+2r^3+r^4)},\frac{3+4r+5r^2+4r^3+3r^4}{3(1+2r+2r^2+2r^3+ 2r^4+ 2r^5 +r^6)}, \ldots,\ \text{etc.} $
The numerators are built according to the pyramid (the pattern is in the diagonals):
1 2 2 2 3 4 5 4 3 4 6 8 8 8 6 4...
and I hope the pattern of the denominator is clear. $S$ seems to converge to
$\frac{1}{1-r^2}$ as the ratios are built up, and I think I have an outline of a proof.
It's a matter of showing that the sum $(\frac 1n)\sum_{k=1}^n \frac{1}{1+r^2-2r\cos(\pi k/n)} $ converges to the Poisson integral as $n$ gets large (I am using the version shown in eq. 7 from the pdf below with $a =1, f =1$).
http://mechse.illinois.edu/research/dstn/teaching_files2/poissonformula.pdf
I think induction might work to show that for each n the sum corresponds to successive elements of S. That part is giving me trouble.