  Even though Zaremba's conjecture remains open, Bourgain and Kontorovich
solved the problem for a full density subset. Nevertheless, there are only a
handful of explicit sequences known to satisfy the strong version of the
conjecture, all of which were obtained using essentially the same algorithm. In
this note, we provide a refined algorithm using the folding lemma for continued
fractions, which both generalizes and improves on the old one. As a result, we
uncover new examples that fulfill the strong version of Zaremba's conjecture.
