A method for deterining the mod-$3^k$ behaviour of recursive sequences

TL;DR

This paper introduces a novel method to determine the behavior of various recursive sequences modulo powers of 3, extending known results and applying to numerous combinatorial and algebraic sequences.

Contribution

The paper presents a new general technique for deriving congruences modulo powers of 3 for recursive sequences with polynomial coefficients, expanding the scope of known results.

Findings

Derived new congruences for Catalan, Motzkin, and Schröder numbers.

Extended known results to higher powers of 3 for multiple sequences.

Applied the method to functions counting free subgroups in modular groups.

Abstract

We present a method for obtaining congruences modulo powers of 3 for sequences given by recurrences of finite depth with polynomial coefficients. We apply this method to Catalan numbers, Motzkin numbers, Riordan numbers, Schr\"oder numbers, Eulerian numbers, trinomial coefficients, Delannoy numbers, and to functions counting free subgroups of finite index in the inhomogeneous modular group and its lifts. This leads to numerous new results, including many extensions of known results to higher powers of 3.