Consider the $2N\times 2N$ matrix

$$A=\begin{pmatrix} a &1 &0&0&0&\ldots&0&1 \\1 &-a&1 & 0 &0 & \ldots & 0&0 \\0 &1&a&1&0 &\cdots &0&0 \\ 0&0&1&-a &1 & \ldots &0&0 \\& & & \cdots \\ 1&0 &0&0&0&\ldots &1&-a\end{pmatrix}$$

Hopefully the structure is clear, but if not I can clarify further.

I am trying to find the eigenvalues of $A$ analytically.

There is a lot of literature exclusively on eigenvalues of tridiagonal matriA$is is neither exactly circulant nor is it exactly tridaigonal. However it is very close to being both.

I have worked out a few cases:

For $N=2$, the eigenvalues are

$$\lambda_{1,2} = \pm a$$ $$\lambda_{3,4} = \pm \sqrt{a^2+4}$$

For $N = 3$, the eigenvalues are

$$\lambda_{1,2} = -\sqrt{1+a^2}$$ $$\lambda_{3,4} = \sqrt{1+a^2}$$ $$\lambda_{5,6} = \pm \sqrt{a^2+4}$$

So it seems there is some sort of 'pattern'.

Any ideas on how I would advance?