[math-fun] seven

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.) Jim Propp