Determinants of some pentadiagonal matrices

In this paper we consider pentadiagonal $(n+1)\times (n+1)$ matrices with two subdiagonals and two superdiagonals at distances $k$ and $2k$ from the main diagonal where $1\le k<2k\le n$. We give an explicit formula for their determinants and also consider the Toeplitz and “imperfect” Toeplitz versions of such matrices. Imperfectness means that the first and last $k$ elements of the main diagonal differ from the elements in the middle. Using the rearrangement due to Egerv'ary and Sz'asz we also show how these determinants can be factorized.