During my research work related to the convergence of estimates, I needed to calculate the eigenvalues of the following symmetric matrix.
$$\Sigma_{n\times n} := \begin{pmatrix} 1 & \frac{a}{2} & \frac{a^2}{2} & \frac{a^4}{2} &\cdots & \frac{a^{n-2}}{2} & \frac{a^{n-1}}{2} \\ \frac{a}{2} & 1 & \frac{a}{2} & \frac{a^2}{2} & \cdots & \frac{a^{n-3}}{2} & \frac{a^{n-2}}{2} \\ \frac{a^2}{2} & \frac{a}{2} & 1 & \frac{a}{2} & \cdots & \frac{a^{n-4}}{2} & \frac{a^{n-3}}{2} \\ \vdots & \vdots & \ddots &\ddots & \cdots & \vdots & \vdots \\ \frac{a^{n-1}}{2} & \cdots & \cdots & \cdots &\cdots & \frac{a}{2}& 1 \end{pmatrix} $$
I tried to guess the answer using analysis on small matrices, but even with a $3 \times 3$ matrix it is difficult to cope. For $2 \times 2$ matrices, everything is obvious. The recurrence has not been compiled.
