Lyness cycles

Abstract

In 1942, R. C. Lyness noted that some recurrence relations generate cycles, irrespective of the initial values. For example, the order 2 recurrence relationgenerates a cycle of period 5 for almost all values of u1 and u2 [1].The globally periodic nature of sequences generated by this recurrence relation can be seen by setting u1 = x and u2 = y. The sequence is thenLyness gave other examples of such recurrence relations but had been unable to find one with period 7 and challenged readers of the Gazette to find such a recurrence relation or prove it to be impossible.No answer to this challenge was forthcoming. However, since Lyness's time, interest in these cycles has been maintained due to links with cross-ratios and elliptic curves. In recent years, Jonny Griffiths has done much to popularise these cycles [2].