# Free subgroup numbers modulo prime powers: the non-periodic case

### (22 pages)

**Abstract.**
In [*J. Algebra* **452** (2016), 372-389],
we characterise when the sequence of free subgroup numbers of a
finitely generated virtually free group *\Gamma*
is ultimately periodic modulo a given prime power.
Here, we show that, in the remaining cases in which the sequence of
free subgroup numbers is not ultimately periodic modulo a given
prime power, the number of free subgroups of index *\lambda*
in *\Gamma* is - essentially -
congruent to a binomial coefficient times a rational function
in *\lambda* modulo a power of a prime that divides a certain invariant
of the group *\Gamma*,
respectively to a binomial sum involving such numbers.
These results allow for a much more efficient computation of
congruences for free subgroup numbers in these cases compared
to the direct recursive computation of these numbers implied
by the generating function results in [*J. London Math. Soc.* (2)
**44** (1991), 75-94].

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