Polynomially-recursive sequences generally have a periodic behavior mod $m$. In this paper, we analyze the period mod $m$ of a second order polynomially-recursive sequence. The problem originally comes from an enumeration of avoiding pattern permutations and appears to be linked with nice number theory notions (the Carmichael function, Wieferich primes, algebraic integers). We give the mod $a^k$ supercongruences, and generalize these results to a class of recurrences.