In this paper we study cluster algebras \myAA of type A2(1). We solve
the recurrence relations among the cluster variables (which form a T--system of
type A2(1)). We solve the recurrence relations among the coefficients of
\myAA (which form a Y--system of type A2(1)). In \myAA there is a
natural notion of positivity. We find linear bases \BB of \myAA such that
positive linear combinations of elements of \BB coincide with the cone of
positive elements. We call these bases \emph{atomic bases} of \myAA. 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 \BB are parameterized
by \ZZ^3 via their g--vectors in every cluster. We prove that the
denominator vector map in every acyclic seed of \myAA restricts to a
bijection between \BB and \ZZ^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 \myAA as linear combinations of elements of
\BB.Comment: Latex, 40 pages; Published online in Algebras and Representation
Theory, springer, 201
