Int. J. Contemp. Math. Sci., Vol. 1, 2006, no. 10, 475-480 On the Recursive Sequence x n+1 = x n−3 1+x n−1 Da˘ gıstan Simsek Selcuk University, Engineering-Architecture Faculty Department of Industrial Engineering 42075 Kamp¨ us/Konya, Turkey dsimsek@selcuk.edu.tr Cengiz Cinar and Ibrahim Yalcinkaya Mathematics Department, Faculty of Education Selcuk University, 42090, Konya, Turkey ccinar@selcuk.edu.tr, iyalcinkaya@selcuk.edu.tr Abstract. In this paper a solution of the following difference equation was investigated x n+1 = x n−3 1+ x n−1 ,n =0, 1, 2, ... where x −3 ,x −2 ,x −1 ,x 0 ∈ (0, ∞). 1. INTRODUCTION Recently there has been a lot of interest in studying the periodic nature of nonlinear difference equations. For some recent result concerning among other problems, the periodic nature of scalar nonlinear difference equations see, for examples [1,2,4,5]. In [3] the following problem was posed.Is there a solution of the following difference equation x n+1 = βx n−1 β + x n for n =0, 1, 2, ... where x −1 ,x 0 ,β ∈ (0, ∞) such that x n → 0 as n →∞. In [6] Stevic assumed that β = 1 and solved the following problem