I was investigating a problem and came up with the following symmetric tridiagonal matrix (with zero diagonal elements):

$$ \left(\begin{array}{cccccc} 0 & a & 0 & \ldots & 0 \\ a & 0 & a^2 & & \vdots \\ 0 & a^2 & 0 & \ddots & 0\\ \vdots & & \ddots & \ddots & a^2\\0 & \ldots & 0 & a^2 & 0 \end{array}\right) $$

The dimension of the matrix is $n \times n$. I am looking for a method to find a formula to calculate its eigenvalues (as a function of $n$) or at least to make a simplification. Any advice is appreciated.