  In this paper, we introduce a variation of the factor complexity, called the
$N$-factor complexity, which allows us to characterize the complexity of
sequences on an infinite alphabet. We evaluate precisely the $N$-factor
complexity for the infinite Fibonacci sequence $\mathbf{f}$ given by Zhang, Wen
and Wu [Electron. J. Comb., 24 (2017)]. The $N$-factor complexity of a class of
digit sequences, whose $n$th term is defined to be the number of occurrences of
a given block in the base-$k$ representation of $n$, is also discussed.
