Given a positive integer N and x ∈ [0, 1] \ Q, 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 ω(x, N) and ω(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 ω(x, N)
and ω(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