I wrote:

Does the group of order 168 shed any light on the curious fact that the map (x,y) |-> (y,(y+1)/x) (from C(x,y) to itself) is of order 7?

(Another way of stating this fact is that the sequence x,y,(y+1)/x, ((y+1)/x)+1)/y,... satisfying a(n+1) = (a(n)+1)/a(n-1) is periodic with period 7. This sequence is usually attributed to someone named Lyness, but I don't know why Lyness studied it.)

My mistake; as Dan Asimov pointed out, the sequence has period 5. Here are the true facts I should have asked about (neither of which involves the number seven): 1) The map (x,y) |-> (y,(y+1)/x) is of order 5; 2) The map (x,y,z) |-> (y,z,(y+z+1)/x) is of order 8. I'd like to know if there's a nice picture to explain what's going on with these recurrences. (But I don't expect it to involve the simple group of order 168!) Jim Propp