 Solution

A monomial of weighted degree $n$ is $x^{n-2j}{y}^j$ with $0\le j\le \lfloor n/2\rfloor$. ThereforeThis is a quasipolynomial of period two, not eventually one polynomial: it equals $n/2+1$ for even $n$ and $(n+1)/2$ for odd $n$. The usual eventual-polynomial theorem for a graded algebra assumes a standard graded algebra, generated in degree one. Here the generator $y$ has degree two, so there is no contradiction.