$A$ is a tridiagonal matrix of the form $$\left( \begin{array}{ccccccccc}d_1&a&0&0&\cdots&0&0&0&0\\a&d_2&a&0&\cdots&0&0&0&0\\0&a&d_3&a&\cdots&0&0&0&0\\0&0&a&d_4&\cdots&0&0&0&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&0&0&\cdots&d_{n-3}&a&0&0\\ 0&0&0&0&\cdots&a&d_{n-2}&a&0\\ 0&0&0&0&\cdots&0&a&d_{n-1}&a\\ 0&0&0&0&\cdots&0&0&a&d_{n} \end{array}\right)$$
and $B$ is another symmetric matrix of the form $$\left(\begin{array}{ccccccccc}0&0&0&0&\cdots&0&0&0&a\\0&0&0&0&\cdots&0&0&0&0\\0&0&0&0&\cdots&0&0&0&0\\0&0&0&0&\cdots&0&0&a&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&0&0&\cdots&0&0&0&0\\ 0&0&0&0&\cdots&0&0&0&0\\ 0&0&0&a&\cdots&0&0&0&0\\ a&0&0&0&\cdots&0&0&0&0 \end{array}\right)\cdot$$
Non-zero entries of $B$ are at the following places:
$B(1,n)=B(n,1)=a,\\B(4,n-1)=B(n-1,4)=a,\\B(5,n-4)=B(n-4,5)=a,\\B(8,n-5)=B(n-5,8)=a,\\B(9,n-8)=B(n-8,9)=a,\\\cdots$
Note that $B^2$ is the diagonal matrix $\mbox{diag}(a^2,0,0,a^2,a^2,0,0,a^2,a^2,\ldots,a^2)$.
What would be an efficient way to find the eigenvalues of $A+B$?