Cluster algebras arising from cluster tubes

We study the cluster algebras arising from cluster tubes with rank bigger than 1. Cluster tubes are 2-Calabi-Yau triangulated categories that contain no cluster tilting objects, but maximal rigid objects. Fix a certain maximal rigid object T in the cluster tube C-n of rank n. For any indecomposable rigid object M in C-n, we define an analogous X-M of Caldero-Chapoton's formula (or Palu's cluster character formula) by using the geometric information of M. We show that X-M, X-M' satisfy the mutation formula when M, M' form an exchange pair, and that X? : M (sic). X-M gives a bijection from the set of indecomposable rigid objects in C-n to the set of cluster variables of cluster algebra of type Cn-1, which induces a bijection between the set of basic maximal rigid objects in C-n and the set of clusters. This yields a surprising result proved recently by Buan-Marsh-Vatne that the combinatorics of maximal rigid objects in the cluster tube C-n encodes the combinatorics of the cluster algebra of type Bn-1, since the combinatorics of cluster algebras of type Bn-1 and of type Cn-1 is the same by a result of Fomin and Zelevinsky. As a consequence, we give a categorification of cluster algebras of type C.