  Motivated by a construction in the theory of cluster algebras (Fomin and
Zelevinsky), one associates to each acyclic directed graph a family of
sequences of natural integers, one for each vertex; this construction is called
a {\em frieze}; these sequences are given by nonlinear recursions (with
division), and the fact that they are integers is a consequence of the Laurent
phenomenon of Fomin and Zelevinsky. If the sequences satisfy a linear recursion
with constant coefficients, then the graph must be a Dynkin diagram or an
extended Dynkin diagram, with an acyclic orientation. The converse also holds:
the sequences of the frieze associated to an oriented Dynkin or Euclidean
diagram satisfy linear recursions, and are even $\mathbb N$-rational. One uses
in the proof objects called $SL_2$-{\em tilings of the plane}, which are
fillings of the discrete plane such that each adjacent 2 by 2 minor is equal to
1. These objects, which have applications in the theory of cluster algebras,
are interesting for themselves. Some problems, conjectures and exercises are
given.
