Cluster Mutation-Periodic Quivers and Associated Laurent Sequences

We consider quivers/skew-symmetric matrices under the action of mutation (in the cluster algebra sense). We classify those which are isomorphic to their own mutation via a cycle permuting all the vertices, and give families of quivers which have higher periodicity. The periodicity means that sequences given by recurrence relations arise in a natural way from the associated cluster algebras. We present a number of interesting, new nonlinear recurrences, necessarily with the Laurent property, of both the real line and the plane. In particular, we show that some of these recurrences can be linearised and, with certain initial conditions, give integer sequences which contain all solutions of some particular Pell equations.