On the structure of partition which the difference of their representation function is a constant

Let N be the set of nonnegative integers. For any set A subset of N, let R-1(A, n), R-2(A, n) and R-3(A, n) be the number of representations of n as n = a+a', a, a' is an element of. A; n = a+a', a, a' is an element of A, a < a'; n = a + a', a, a' is an element of A, a <= a' respectively. In this paper, for each i is an element of {2, 3}, we determine the structure of the partition N = A boolean OR B with A boolean AND B = empty set such that R-i (A, n) - R-i (B, n) = k holds for all sufficiently large integers n, where k is a given integer.