In this paper we study the behavior of the difference equation {aastex} {amsbsy} {amsfonts} {amssymb} {bm} {mathrsfs} {pifont} {stmaryrd} {textcomp} {upgreek} {portland,xspace} {amsmath,amsxtra} {bbm} {empty} {10}{9}{7}{6} {document} xn + 1 = axn - 2 + bxₙ xn - 2 cxₙ + dxn - 3 ,n = 0,1,... {document} where the initial conditions x−3 , x−2 , x−1 , x0 are arbitrary positive real numbers and a, b, c, d are positive constants. Also, we give the solution of some special cases of this equation.
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E. M. Elsayed (2009) studied this question.