The determinants of certain \((0,1)\) Toeplitz matrices
This paper deals with pentadiagonal Toeplitz matrices, namely square Toeplitz matrices with four subdiagonals, two above and two below the main diagonal, that are nonzero (see [\textit{M. Andelić} and \textit{C. M. da Fonseca}, ``Some determinantal considerations for pentadiagonal matrices'', Linear Multilinear Algebra (to appear), \url{doi:10.1080/03081087.2019.1708845}]). The following is the setting of this paper. Consider natural numbers \(n> t \geq 2\) and define the \(n \times n\) matrix \(A(n,t)\) by \[ [A(n,t)]_{ij} = \begin{cases} 1 & \quad\text{if } j-i \in \{-2, -1, 0, 1, t\} \\ 0 &\quad \text{otherwise.} \end{cases} \] Suppose \(\Delta(n,t) = \det A(n,t)\). The main result claims that for any fixed natural number \(k \geq 1\), the sequence \(\Delta(n, n-k)\) has period four on the range \(n \geq 2k\). In particular, this settles a question posed in the above paper.
scientific article; zbMATH DE number 7337633