On the eigenvalues of Toeplitz matrices with two off-diagonals

Consider the Toeplitz matrix $T_n(f)$ generated by the symbol $f(\theta)=\hat{f}_r e^{\mathbf{i}r\theta}+\hat{f}_0+\hat{f}_{-s} e^{-\mathbf{i}s\theta}$, where $\hat{f}_r, \hat{f}_0, \hat{f}_{-s} \in \mathbb{C}$ and $0<r<n,~0<s<n$. For $r=s=1$ we have the classical tridiagonal Toeplitz matrices, for which the eigenvalues and eigenvectors are known. Similarly, the eigendecompositions are known for $1<r=s$, when the generated matrices are ``symmetrically sparse tridiagonal''. In the current paper we study the eigenvalues of $T_n(f)$ for $1\leq r<s$, which are ``non-symmetrically sparse tridiagonal''. We propose an algorithm which constructs one or two ad hoc matrices smaller than $T_n(f)$, whose eigenvalues are sufficient for determining the full spectrum of $T_n(f)$. The algorithm is explained through use of a conjecture for which examples and numerical experiments are reported for supporting it and for clarifying the presentation. Open problems are briefly discussed.