= Solution
A monomial of weighted degree $n$ is $x^{n-2j}y^j$ with $0\leq j\leq\lfloor n/2\rfloor$. Therefore
$$
\boxed{F_S(n)=\dim_kS_n=\left\lfloor\frac n2\right\rfloor+1.}
$$
This 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.
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