On cluster variables of rank two acyclic cluster algebras

In this note, we find an explicit formula for the Laurent expression of cluster variables of coefficient-free rank two cluster algebras associated with the matrix $\left(\begin{array}{ccc}0& c-c& 0\end{array}\right)$, and show that a large number of coefficients are non-negative. As a corollary, we obtain an explicit expression for the Euler-Poincar'{e} characteristics of the corresponding quiver Grassmannians.