Let a Nim game be represented by a sequence of positive integers. We call a Nim of size $n$ when the sum of its elements is $n$.
Let $a(n)$ be the number of Nim games of size $2n$ with Nim sum 0. Similarly, let $b(n)$ be the number of Nim games of size $2n$ with Nim sum 2.
I'm trying to prove (or possibly disprove) $$\lim_{n\to\infty}\frac{a(n)}{b(n)}=1.$$
I've calculated the first values with help of a computer; by $n=250$, this fraction is about $1-10^{-50}$, so there's clearly some convergence going on. I haven't been able to prove it, however. I've been trying to establish some sort of "partial bijection" between the sets of Nim games of Nim sum 0 and 2, but that hasn't worked. I've tried using recursion, I've gotten nothing. Any help?