The finite cosine–Kronecker-delta formula for polygonal mean-distance moments

Let $K$ be a regular $n$-sided polygon with $n\geq 3$, circumscribed by a circle of unit radius. For a positive even integer $p$, let $L_{22}^{(p)}$ denote the corresponding $(2,2)$ moment. Finite cosine–Kronecker-delta formula. There exist numbers $a_{jp}$ and $b_{jp}$ such that

$$L_{22}^{(p)}=\sum_{j=0}^{p/2}\left(a_{jp}\cos\left(\frac{2\pi j}{n}\right)+b_{jp}\delta_{jn}\right).$$

The formula is suggested by explicit calculations for several higher even moments and predicts that every such moment has a finite cosine expansion together with Kronecker-delta corrections at the exceptional values where $j=n$.