  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.
