On the higher order rational recursive sequence xn=Axn-k+Bxn-3k

The main purpose of the current paper is to prove that every positive solution of the delay difference equationxn=Axn-k+Bxn-3k,where A, B ∈ (0, ∞), the initial conditions x−3k+1, x−3k+2, …, x0 ∈ (0, ∞) and k ∈ {1, 2, 3, …}, converges eventually to a period-k solution. We also give the results of computational examples to support our theoretical discussion.