  We define the notion of infinite friezes of positive integers as a variation
of Conway-Coxeter frieze patterns and study their properties. We introduce
useful gluing and cutting operations on infinite friezes. It turns out that
triangulations of once-punctured discs give rise to periodic infinite friezes
having special properties, a notable example being that each diagonal consists
of a collection of arithmetic progressions. Furthermore, we work out a
combinatorial interpretation of the entries of infinite friezes associated to
triangulations of once-punctured discs via matching numbers for certain
combinatorial objects, namely periodic triangulations of strips. Alternatively,
we consider a known algorithm that as we show computes as well these entries.
