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 $n=s_1+s_2$, $s_1,s_2\in S$ and $s_1<s_2$. Let A be the set of all nonnegative integers which contain an even number of digits $1$ in their binary representations and $B=\mathbb {N}\setminus A$. Put $A_l=A\cap [0,2^l-1]$ and $B_l=B\cap [0,2^l-1]$. We prove that if $C \cup D=[0, m]\setminus \{r\}$ with $0<r<m$, $C \cap D=\emptyset $ and $0 \in C$, then $R_{C}(n)=R_{D}(n)$ for any nonnegative integer n if and only if there exists an integer $l \geq 1$ such that $m=2^{l}$, $r=2^{l-1}$, $C=A_{l-1} \cup (2^{l-1}+1+B_{l-1})$ and $D=B_{l-1} \cup (2^{l-1}+1+A_{l-1})$. Kiss and Sándor [‘Partitions of the set of nonnegative integers with the same representation functions’, Discrete Math. 340 (2017), 1154–1161] proved an analogous result when $C\cup D=[0,m]$, $0\in C$ and $C\cap D=\{r\}$.