Thanks to F. Marcellin (Eigenproblems for tridiagonal 2-Toeplitz matrices and quadratic polynomial mappings) and M. J. C. Gover (The Eigenproblem of a Tridiagonal 2-Toeplitz Matrix) there is a close form for the tridiagonal 2-Toeplitz Matrix with form
$$M=\begin{array}{ccccdcc} a1&b1&0 &0&0&...&0 \\ b1&a2&b2&0&0&...&0 \\ 0&b2&a1&b1&0&...&0\\ 0&0&b1&a2&b2&...&0\\ \vdots&\vdots&\vdots&\ddots&\ddots&\ddots&\vdots\\ 0&0&0&0&b2&a1&b1\\ 0&0&0&0&0&b1&a2 \end{array}$$
My question is: what is a close form of eigenvalues of the following matrix
$$M_1=\begin{array}{ccccdcc} a1&b1&0 &0&0&...&0 \\ b1&a1&b2&0&0&...&0 \\ 0&b2&a2&b1&0&...&0\\ 0&0&b1&a2&b2&...&0\\ \vdots&\vdots&\vdots&\ddots&\ddots&\ddots&\vdots\\ 0&0&0&0&b2&a2&b1\\ 0&0&0&0&0&b1&a2 \end{array}$$
where there is a difference between its diagonal elements with the matrix M