Weighted sums of generalized polygonal numbers with coefficients 1 or 2

In this article, we consider weighted sums of generalized polygonal numbers with coefficients $1$ or $2$. We show that for any $m\ge10$, those weighted sums of generalized $m$-gonal numbers represent every non-negative integers if they only represent $1$, $m-4$, and $m-2$. Furthermore, we study representations of sums of four generalized polygonal numbers with coefficients $1$ or $2$.