A remark of Zagier

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 ith term defined by requiring that the ith term multiplied by the i-2th term is 1 plus the ith term. That is, x_{i+1}=(1+x_i)/x_{i-1} or x_{i-1}x_{i+1}=1+x_i.

Let’s start with the values x_1=3 and x_2=4 as Zagier does. The resulting sequence is 3,4,5/3,2/3,1,3,4,5/3,2/3,1,.... Indeed, it is not hard to prove that, for any pair of initial values, the sequence is 5-periodic: x_{i+5}=x_i.

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 A_2, whose cluster algebra has 5 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.

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