In this paper, we consider the nonlinear difference equation x n + 1 = f ( x n − l + 1 , x n − 2 k + 1 ) , n = 0 , 1 , … , where k , l ∈ { 1 , 2 , … } with 2 k ≠ l and gcd ( 2 k , l ) = 1 and the initial values x − α , x − α + 1 , … , x 0 ∈ ( 0 , + ∞ ) with α = max { l − 1 , 2 k − 1 } . We give sufficient conditions under which every positive solution of this equation converges to a ( not necessarily prime ) 2-periodic solution, which extends and includes corresponding results obtained in the recent literature.