  We observe that a sequence satisfies Lucas congruences modulo $p$ if and only
if its values modulo $p$ can be described by a linear $p$-scheme, as introduced
by Rowland and Zeilberger, with a single state. This simple observation
suggests natural generalizations of the notion of Lucas congruences. To
illustrate this point, we prove explicit generalized Lucas congruences for
integer sequences that can be represented as the constant terms of $P(x,y)^n
Q(x,y)$ where $P$ and $Q$ are certain Laurent polynomials.

Number Theory;Combinatorics