Weighted sums of generalized polygonal numbers with coefficients $1$ or $2$

Abstract

We consider weighted sums of generalized polygonal numbers with coefficients $1$ or $2$. We show that for any $m\ge 10$, a weighted sum of generalized $m$-gonal numbers represents every non-negative integer if it represents $1$, $m-4$, and $m-2$. Furtherm