# A
method for determining the mod-3^{k} behaviour of
recursive sequences

### (65 pages)

**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,
Schröder 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.

The following versions are available:

The paper is accompanied by the following *Mathematica* files:
By using the notebook (which requires the other file as input file ),
you are able to
redo (most of) the computations that are presented in
this article.

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