  We introduce a family of maps generating continued fractions where the digit
$1$ in the numerator is replaced cyclically by some given non-negative integers
$(N_1,\ldots,N_m)$. We prove the convergence of the given algorithm, and study
the underlying dynamical system generating such expansions. We prove the
existence of a unique absolutely continuous invariant ergodic measure. In
special cases, we are able to build the natural extension and give an explicit
expression of the invariant measure. For these cases, we formulate a
Doeblin-Lenstra type theorem. For other cases we have a more implicit
expression that we conjecture gives the invariant density. This conjecture is
supported by simulations. For the simulations we use a method that gives us a
smooth approximation in every iteration.
