Eigenvalues for tridiagonal 3-Toeplitz matrices

In this paper, we study the eigenvalues of real tridiagonal 3𝑇oeplitz matricesof different order. When the order of a tridiagonal 3𝑇oeplitz matrix is n = 3𝑘 + 2, the eigenvalues were found explicitly. Here, we consider the distribution of eigenvaluesfor a tridiagonal 3Toeplitz matrix of orders 𝑛 = 3𝑘 and 𝑛 = 3𝑘 + 1. We explain ourmethod by finding roots of a combination of Chebyshev polynomials of the secondkind. This distribution solves the eigenproblem for integer powers of such matrices.