For any positive integers \(k_1, k_2\) and any set \(A \subseteq {\mathbb {N}}\) , let \(R_{k_1,k_2}(A,n)\) be the number of solutions of the equation \(n=k_1a_1+k_2a_2\) with \(a_1,a_2\in A\) . For two integers \(k>1\) and \(t\ge 0\) , define \(g_k(t)\) to be the number of sets \(A\subseteq {\mathbb {N}}\) such that \(R_{1,k}(A,n)-R_{1,k}({\mathbb {N}}\setminus A,n)=1\) holds for all integers \(n\ge t\) . In this paper, we prove that if k, m are two integers with \(k>1\) and \(m\ge 0\) , then there exists a unique set A such that \(R_{1,k}(A,n)-R_{1,k}({\mathbb {N}}\setminus A,n)=m+1\) for all integers \(n\ge mk\) . And we also give the exact formula of \(g_k(t)\) when \(t\le k.\)
