  In the theory of continued fractions, Zaremba's conjecture states that there
is a positive integer $M$ such that each integer is the denominator of a
convergent of an ordinary continued fraction with partial quotients bounded by
$M$. In this paper, to each such $M$ we associate a regular sequence---in the
sense of Allouche and Shallit---and establish various properties and results
concerning the generating function of the regular sequence. In particular, we
determine the minimal algebraic relation concerning the generating function and
its Mahler iterates.
