Some periodic and non-periodic recursions

It is well known that the recurrence relations x(n) = 1/x(n-1), x(n) = 1 + x(n-1)/x(n-2) and x(n) = 1 + x(n-1) + x(n-2)/x( n-3) are periodic, in the sense that they define periodic sequences for all choi ces of the initial data, and lead to sequences with periods 2, 5 and 8, res pectively. In this paper we determine all periodic recursions of the form x(n) = a(0) + a(1)x(n-1) + ... +a(h)x(n-k)/x(n-k-1) where a(0),...,a(k) are complex numbers, a(1),...,a(k-1) are non-zens and a (k) = 1. We find that, apart from the three recursions listed above, only x(n) = x(n-1)/x(n-2) and x(n) = -1 - x(n-1) + x(n-2)/x(n-3) lead to periodic sequences (with periods 6 and 8). The non-periodicity of ( R) when k greater than or equal to 3 (or k = 2 and a(0) = 0) depends on the connection between (R) and the recurrence relations x(n) = max(x(n-1),...,x(n-k)) - x(n-k-1) and x(n) = max(x(n-1),...,x(n-k),0) - x(n-k-1) We investigate these recursions together with the related x(n) = /max(x(n-1),...,x(n-k))/ - x(n-k-1) Each of(A), (B), and (C) leads to periodic sequences if k = 1 (with periods 6, 5, and 9, respectively). Also, for k = 2, (B) leads to periodicity with period 8. However, no other cases give rise to periodicity. We also prove that every real sequence satisfying any of (A), (B), and (C) must be bounde d. As a consequence, we find that for an arbitrary k, every rational sequen ce satisfying any of (A, (B), and (C) must be periodic. 2000 Mathematics Su bject Classification.