Friezes and a construction of the Euclidean cluster variables

Let Q be a Euclidean quiver. Using friezes in the sense of Assem-Reutenauer-Smith, we provide an algorithm for computing the (canonical) cluster character associated with any object in the cluster category of Q. In particular, this algorithm allows us to compute all the cluster variables in the cluster algebra associated with Q. It also allows us to compute the sum of the Euler characteristics of the quiver Grassmannians of any module M over the path algebra of Q.