Consider the following matrix: $$L(P_5):=\left[\begin{array}{@{}rrrrr@{}} 1&-1& 0& 0& 0\\ -1& 2&-1& 0& 0\\ 0&-1& 2&-1& 0\\ 0& 0&-1& 2&-1\\ 0& 0& 0&-1& 1 \end{array}\right].$$ Its eigendecomposition is well known: its eigenvectors are the columns of the type-II Discrete Cosine Transform, up to a transposition; and its eigenvalues are the roots of a type-II Chebyschyov polynomial after certain linear transformation, and zero.
Now take a look at these matrices: $$L(\mathcal{D}_6):=\left[\begin{array}{@{}rrrrrr@{}} 2&-1&-1& 0& 0& 0\\ -1& 3&-1&-1& 0& 0\\ -1&-1& 4&-1&-1& 0\\ 0&-1&-1& 4&-1&-1\\ 0& 0&-1&-1& 3&-1\\ 0& 0& 0&-1&-1& 2 \end{array}\right],$$ $$L(\mathcal{M}_8):=\left[\begin{array}{@{}rrrrrrrr@{}} 3&-1&-1&-1& 0& 0& 0& 0\\ -1& 4&-1&-1&-1& 0& 0& 0\\ -1&-1& 5&-1&-1&-1& 0& 0\\ -1&-1&-1& 6&-1&-1&-1& 0\\ 0&-1&-1&-1& 6&-1&-1&-1\\ 0& 0&-1&-1&-1& 5&-1&-1\\ 0& 0& 0&-1&-1&-1& 4&-1\\ 0& 0& 0& 0&-1&-1&-1& 3 \end{array}\right],$$ and so on. Are there closed-form expresions for the eigendecomposition of these new matrices? If so, I am curious whether the eigendecomposition can be escalated to higher order matrices (hence the subscripts.)