  Let V be an infinite matrix with rows and columns indexed by the positive
integers, and entries in a field F. Suppose that v_{i,j} only depends on i-j
and is 0 for |i-j| large. Then V^n is defined for all n, and one has a
"generating function" G=\sum a_{1,1}(V^n)z^n. Ira Gessel has shown that G is
algebraic over F(z). We extend his result, allowing v_{i,j} for fixed i-j to be
eventually periodic in i rather than constant. This result and some variants of
it that we prove will have applications to Hilbert-Kunz theory.
