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+1−1yn+yn+1andzn+2=(zn+1+zn)modπ. 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+axn+xn−2and xn+1=xn−1xn−2+axn−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].