In Evan and Hendel's recent proof of an outstanding conjecture on the resistance distances of a family of linear 3-trees, a key technique in the proof was calculating the recursion satisfied by a family of determinants. The underlying algorithm employed to prove the conjecture converged (i.e. terminated) in the particular case studied, and the paper presented an open question on when such a procedure converges in general. This paper proves convergence of the procedure for an arbitrary family of determinants of banded, square, Toeplitz matrices. Moreover, the algorithm in this paper improves several aspects of the algorithm of Evans and Hendel.