On the monotonicity properties of additive representation functions, II

For a set A of positive integers and any positive integer n , let R 1 ( A , n ) , R 2 ( A , n ) and R 3 ( A , n ) denote the number of solutions of a + a ′ = n with the additional restriction a , a ′ ∈ A ; a , a ′ ∈ A , a < a ′ and a , a ′ ∈ A , a ≤ a ′ respectively. In this paper, we specially focus on the monotonicity of R 3 ( A , n ) . Moreover, we show that there does not exist any set A ⊂ N such that R 2 ( A , n ) or R 3 ( A , n ) is eventually strictly increasing.