The Fibonacci sequence modulo π, chaos and some rational recursive equations

In this paper, we provide the closed form solution to the inter-related equations yn+2 = ynyn+1 − 1 yn + yn+1 and zn+2 = (zn+1 +zn) mod π. Both of these equations were suggested as open problems in the book by Kocic and Ladas [V.L. Kocic, G. Ladas, Global Behavior of Nonlinear Difference Equations of High Order with Applications, Kluwer Academic, Dordrecht, 1993]. We also give the closed form solution to the equations xn+1 = xnxn−2 + a xn +xn−2 and xn+1 = xn−1xn−2 + a xn−1 +xn−2 , studied by X. Li and D. Zhu [X. Li, D. Zhu, Two rational recursive sequences, Comput. Math. Appl. 47 (2004) 1487–1494].  2005 Elsevier Inc. All rights reserved