# 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 $i$th term defined by requiring that the $i$th term multiplied by the $i-2$th term is $1$ plus the $i$th 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.