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.