  We prove that for arbitrary partitions $\mathbf{\lambda} \subseteq
\mathbf{\kappa},$ and integers $0\leq c<r\leq n,$ the sequence of Schur
polynomials $S_{(\mathbf{\kappa} + k\cdot \mathbf{1}^c)/(\mathbf{\lambda} +
k\cdot \mathbf{1}^r)}(x_1,...,x_n)$ for $k$ sufficiently large, satisfy a
linear recurrence. The roots of the characteristic equation are given
explicitly. These recurrences are also valid for certain sequences of minors of
banded Toeplitz matrices.
  In addition, we show that Widom's determinant formula from 1958 is a special
case of a well-known identity for Schur polynomials.
