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Trigonometric sum and residues
I am interested in the sum
$$
\sum_{n=1}^k 2\biggl[\sin\biggl(\frac{n\pi}{2k+2}\biggr)\biggr]^{-2g}
$$
where $k$, $g$ are integers. It is not too hard to show that this can also be expressed as
$$
-1-...
- 195