Matrices of this form are called "periodic Jacobi matrices", see for example, The Construction of Jacobi and Periodic Jacobi Matrices With Prescribed Spectra. If you search for this name, you will find a very extensive literature; there are also MO questions that address the spectrum of periodic Jacobi matrices, for example, here.

Matrices of this form are called "periodic Jacobi matrices", see for example,

-- The spectrum of Jacobi matrices.

-- The construction of Jacobi and periodic Jacobi matrices with prescribed spectra.

-- Continued fractions and periodic Jacobi matrices.