Many integer sequences including the Catalan numbers, Motzkin numbers, and the Apéry numbers can be expressed as the constant term of PⁿQ. for Laurent polynomials P and Q. These are often called constant term sequences. In this paper, we characterize the prime powers for which sequences of this form modulo p^a, and others built from these sequences, are uniformly recurrent. For all other prime powers, we show that the frequency of 0 is 1. We accomplish this by introducing a novel linear representation of constant term sequences modulo p^a, which is of independent interest.