p-linear schemes for sequences modulo pr

Many interesting combinatorial sequences, such as Apéry numbers and Franel numbers, enjoy the so-called Lucas property modulo almost all primes p. Modulo prime powers pr such sequences have a more complicated behaviour which can be described by matrix versions of the Lucas property called p-linear schemes. They are generalizations of finite p-automata. In this paper we construct such p-linear schemes and give upper bounds for the number of states which, for fixed r, do not depend on p.