Suppose I have the symmetric tridiagonal matrix:

$ \begin{pmatrix} a & b_{1} & 0 & ... & 0 \\\ b_{1} & a & b_{2} & & ... \\\ 0 & b_{2} & a & ... & 0 \\\ ... & & ... & & b_{n-1} \\\ 0 & ... & 0 & b_{n-1} & a \end{pmatrix} $

All of the entries can be taken to be positive real numbers and all of the $a_{i}$ are equal. I know that when the $b_{i}$'s are equal (the matrix is uniform), there are closed-form expressions for the eigenvalues and eigenvectors in terms of cosine and sine functions. Additionally, I know of the recurrence relation:

$det(A_{n}) = a\cdot det(A_{n-1}) - b_{n-1}^{2}\cdot det(A_{n-2})$

Additionally, since my matrix is real-symmetric, I know that its eigenvalues are real.

Is there anything else I can determine about the eigenvalues? Furthermore, is there a closed-form expression for them?