Partitions of Natural Numbers and Their Representation Functions

For a given set A of nonnegative integers the representation functions R₂(A, n), R₃(A, n) are defined as the number of solutions of the equation n = x+y, x, y ∈ A with condition x <y,  x \leq y respectively. In this note we are going to determine the partitions of natural numbers into two parts such that their representation functions are the same from a certain point onwards.