Abstract Let N be the set of all nonnegative integers. For a set A N, let R₂ (A, n) and R₃ (A, n) be the number of solutions of the equation n=a₁+a₂ with a₁<a₂, a₁, a₂ A and with a₁ a₂, a₁, a₂ A, respectively. If -N g N, Yan ‘On the structure of partition which the difference of their representation function is a constant’, Period. Math. Hungar. 82 (2021), 149–152 showed that there is a set A N such that Rᵢ (A, n) -Rᵢ (N A, n) =g for all integers n 2N-1, where N is a positive integer. In this paper, we prove that if g₁, g₂ are nonnegative integers with g₁ g₂, then there does not exist A N such that Rᵢ (A, 2n) -Rᵢ (N A, 2n) =g₁ and Rᵢ (A, 2n+1) -Rᵢ (N A, 2n+1) =g₂ for all sufficiently large integers n.
Yan et al. (Thu,) studied this question.