  Let $\mathbb{N}$ be the set of all nonnegative integers. For $S\subseteq
\mathbb{N}$ and $n\in \mathbb{N}$, let $R_S(n)$ denote the number of solutions
of the equation $n=s+s'$, $s, s'\in S$, $s<s'$.
  In this paper, we determine the structure of all sets $A$ and $B$ such that
$A\cup B=\mathbb{N}\setminus\{r+mk:k\in\mathbb{N}\}$, $A\cap B=\emptyset$ and
$R_{A}(n)=R_{B}(n)$ for every positive integer $n$, where $m$ and $r$ are two
integers with $m\ge 2$ and $r\ge 0$.
