  In a Coxeter group $W$, an element is fully commutative if any two of its
reduced expressions can be linked by a series of commutation of adjacent
letters. These elements have particularly nice combinatorial properties, and
also index a basis of the generalized Temperley--Lieb algebra attached to $W$.
  We give two results about the sequence counting these elements with respect
to their Coxeter length. First we prove that it always satisfies a linear
recurrence with constant coefficients, by showing that reduced expressions of
fully commutative elements form a regular language. Then we classify those
groups $W$ for which the sequence is ultimately periodic, extending a result of
Stembridge. These results are applied to the growth of generalized
Temperley--Lieb algebras.
