Circulant matrices are famous because they are diagonalized by vectors being the basis functions to the Discrete Fourier Transform.
But what happens if we slightly modify a circulant matrix, so that there is a "gap" of $2$ instead of $1$ for each new row? Let us only consider odd sized matrices as AndreasH pointed out degeneracy of even sized ones below.
For example like this: $$D = \begin{bmatrix}1&-1&0&0&0&0&0\\ 0& 0& 1& -1& 0& 0&0\\ 0& 0& 0& 0& 1& -1&0\\ -1& 0& 0& 0& 0& 0&1\\ 0& 1& -1& 0& 0& 0&0\\ 0& 0& 0& 1& -1& 0&0\\ 0& 0& 0& 0& 0& 1&-1 \end{bmatrix}$$
Can we derive as a function of say the first row what the eigenbasis shall look like in a general case?