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

-- <A HREF="http://link.springer.com/article/10.1007%2FBF01418827">The spectrum of Jacobi matrices.</A>

-- <A HREF="http://www.jstor.org/stable/2006386">The construction of Jacobi and periodic Jacobi matrices with prescribed spectra</A>.

-- <A HREF="http://www.sciencedirect.com/science/article/pii/002437959290008X#">Continued fractions and periodic Jacobi matrices.</A>