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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 101 / 1 / c / Solution

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A quasipolynomial is a function $f:\mathbb Z\to\mathbb C$ for which there are <polynomial>[polynomials] $P_0,\ldots,P_{m-1}$ such that $f(n)=P_r(n)$ whenever $n\equiv r\pmod m$. The least possible positive $m$ is its period.

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