In this paper, we investigate the boundedness, invariant interval, semicycle and global attractivity of all positive solutions of the equation x n+1 = α+γ x n−1 A+Bx n +C x n−1 , n = 0, 1, . . . , where the parameters α, γ , A, B, C ∈ (0, ∞) and the initial conditions y −1 , y 0 are nonnegative real numbers. We show that if the equation has no prime period-two solutions, then the positive equilibrium of the equation is globally asymptotically stable. Our results solve partially the conjecture proposed by Kulenović and Ladas in their monograph