Last week, I wrote (in response to Dan Hoey's posting):
My mistake; as Dan Asimov pointed out, the sequence has period 5.
(thereby showing that I can confuse one Dan with another with as much facility as I can confuse small natural numbers :-) ). I should mention that the theory of cluster algebras currently being developed by Fomin and Zelevinsky gives a context for these facts, but it doesn't really explain them in an elementary way. Here is a restatement of the "order-5 fact", along with two other facts of a similar kind: 1) Any sequence x(1),x(2),x(3),... defined by the recurrence x(n) = (x(n-1)+1)/x(n-2) (for n>2) has period 5. 2) Any sequence x(1),x(2),x(3),... defined by the recurrence { (x(n-1)+1)/x(n-2) (for n>2, n odd) x(n) = { { (x(n-1)^2+1)/x(n-2) (for n>2, n even) has period 6. 3) Any sequence x(1),x(2),x(3),... defined by the recurrence { (x(n-1)+1)/x(n-2) (for n>2, n odd) x(n) = { { (x(n-1)^3+1)/x(n-2) (for n>2, n even) has period 8. (Note that this is different from the period-8 recurrence I mentioned in my previous email.) In the theory of cluster algebras, these recurrences correspond to the Lie algebras A_2; B_2 (or C_2); and G_2 (respectively). But I'm just quoting someone else here; I don't actually understand the link between Lie algebras and cluster algebras. Nor do I have the foggiest notion of whether the x(n) = (x(n-1)+x(n-2)+1)/x(n-3) recurrence fits into the story somehow. But since there's such a nice story for the fact that the map t -> 1-1/t has order 3, it doesn't seem far-fetched to hope for something similar in these cases. Jim Propp