In uniform distribution theory, discrepancy is a quantitative measure for the irregularity of distribution of a sequence modulo one. At the moment the concept of digital (t,s)-sequences as introduced by Niederreiter provides the most powerful constructions of s-dimensional sequences with low discrepancy. In one dimension, recently Faure proved exact formulas for different notions of discrepancy for the subclass of NUT digital (0,1)-sequences. It is the aim of this paper to generalize the concept of NUT digital (0,1)-sequences and to show in which sense Faure's formulas remain valid for this generalization. As an application we obtain best possible lower bounds for the star discrepancy of several subclasses of (0,1)-sequences.