On the weighted sum of consecutive values of an additive representation function

Let $\mathbb{N}$ be the set of nonnegative integers. For any set $A\subset\mathbb{N}$, let $R_{A}(n)$ denote the number of solutions of the equation $n=a+b$ with $a,b\in A$. Recently, Kiss and Sándor established some relations between $|\lambda_0R_A(n)+\lambda_1R_A(n-1)+\cdots+\lambda_dR_A(n-d)|$ and $|\{m:m\le n,\lambda_0\chi_A(m)+\lambda_1\chi_A(m-1)+\cdots+\lambda_d\chi_A(m-d)\not=0\}|$, where $\chi_A(k)=1$ if $k\in A$, otherwise $\chi_A(k)=0$. In this paper, we improve one of the results of Kiss and Sándor to the best possible up to a constant factor.