  Given a positive integer $N$ and $x$ irrational between zero and one, an
$N$-continued fraction expansion of $x$ is defined analogously to the classical
continued fraction expansion, but with the numerators being all equal to $N$.
Inspired by Sturmian sequences, we introduce the $N$-continued fraction
sequences $\omega(x,N)$ and $\hat{\omega}(x,N)$, which are related to the
$N$-continued fraction expansion of $x$. They are infinite words over a two
letter alphabet obtained as the limit of a directive sequence of certain
substitutions, hence they are $S$-adic sequences. When $N=1$, we are in the
case of the classical continued fraction algorithm, and obtain the well-known
Sturmian sequences. We show that $\omega(x,N)$ and $\hat{\omega}(x,N)$ are
$C$-balanced for some explicit values of $C$ and compute their factor
complexity function. We also obtain uniform word frequencies and deduce unique
ergodicity of the associated subshifts. Finally, we provide a Farey-like map
for $N$-continued fraction expansions, which provides an additive version of
$N$-continued fractions, for which we prove ergodicity and give the invariant
measure explicitly.
