Some Periodic Solutions of the Lyness Equation

Theorem: For any real number x >1 and any natural number i congruent to 2, 4, 5, 6, 7

or 9 modulo 12, there is a positive value of a such that equation (1 ), ~~ = ~~ , has a periodic solution of length 2i + 1 with x_1 = x and Xo = x - 1.