  In this expository article we collect the integer sequences that count
several different types of matrices over finite fields and provide references
to the Online Encyclopedia of Integer Sequences (OEIS). Section 1 contains the
sequences, their generating functions, and examples. Section 2 contains the
proofs of the formulas for the coefficients and the generating functions of
those sequences if the proofs are not easily available in the literature. The
cycle index for matrices is an essential ingredient in most of the derivations.
