Matrices of this form are called "periodic Jacobi matrices", see for example, <A HREF="http://www.jstor.org/stable/2006386">The Construction of Jacobi and Periodic Jacobi Matrices With Prescribed Spectra</A>. 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, <A HREF="http://mathoverflow.net/questions/65657/largest-eigenvalue-of-a-periodic-jacobi-matrix">here.</A>