This is a short advertisement for cluster algebras via a remark of Don Zagier that was quoted on a Stack.Exchange post I came across this evening. Zagier invites one to consider a sequence with the first two values known and then with the th term defined by requiring that the th term multiplied by the th term is plus the th term. That is, or .
Let’s start with the values and as Zagier does. The resulting sequence is . Indeed, it is not hard to prove that, for any pair of initial values, the sequence is 5-periodic: .
The recurrence defining the sequence is a trinomial relation that one might hope to capture in mutations of a quiver; in other words, in a cluster algebra. This is indeed possible as I discuss. The associated quiver is an orientation of the Dynkin diagram , whose cluster algebra has seeds as part of the classification for finite-type cluster algebras. This is another way of seeing the periodicity or finiteness of the situation as stemming from a quiver with strong finiteness properties. Advertisement over.