Ramus's Identity

Theorem
Let $k, m, n \in \Z_{\ge 0}$ be positive integers such that $0 \le k < m$.

Then:


 * $\displaystyle \dbinom n k + \dbinom n {m + k} + \dbinom n {2 m + k} + \cdots = \dfrac 1 m \sum_{0 \mathop \le j \mathop < m} \left({2 \cos \dfrac {j \pi} m}\right)^n \cos \dfrac {j \left({n - 2 k}\right) \pi} m$