  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 $2$.
