Eigenvalues for tridiagonal 3-Toeplitz matrices

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