  Frieze patterns, as introduced by Coxeter in the 1970's, are closely related
to cluster algebras without coefficients. A suitable generalization of frieze
patterns, linked to cluster algebras with coefficients, has only briefly
appeared in an unpublished manuscript by Propp. In this paper we study these
frieze patterns with coefficients systematically and prove various fundamental
results, generalizing classic results for frieze patterns. As a consequence we
see how frieze patterns with coefficients can be obtained from classic frieze
patterns by cutting out subpolygons from the triangulated polygons associated
to classic Conway-Coxeter frieze patterns. We address the question of which
frieze patterns with coefficients can be obtained in this way and solve this
problem completely for triangles. Finally, we prove a finiteness result for
frieze patterns with coefficients by showing that for a given boundary sequence
there are only finitely many (non-zero) frieze patterns with coefficients with
entries in a discrete subset of the complex numbers.
