In this paper we study cluster algebras of type A2(1). We solve the recurrence relations among the cluster variables (which form a T--system of type A2(1)
Nearby in the stack
). We solve the recurrence relations among the coefficients of
(which form a Y--system of type
A2(1)
). In
there is a natural notion of positivity. We find linear bases
of
such that positive linear combinations of elements of
coincide with the cone of positive elements. We call these bases atomic bases of
. These are the analogue of the "canonical bases" found by Sherman and Zelevinsky in type
A1(1)
. Every atomic basis consists of cluster monomials together with extra elements. We provide explicit expressions for the elements of such bases in every cluster. We prove that the elements of
are parameterized by
³
via their
g
--vectors in every cluster. We prove that the denominator vector map in every acyclic seed of
restricts to a bijection between
and
³
. In particular this gives an explicit algorithm to determine the "virtual" canonical decomposition of every element of the root lattice of type
A2(1)
. We find explicit recurrence relations to express every element of