Almost Universal Weighted Ternary Sums of Polygonal Numbers

For a natural number $m$, generalized $m$-gonal numbers are defined by the formula $p_m(x)=\frac{(m-2)x^2-(m-4)x}{2}$ with $x\in \mathbb{Z}$. In this paper, we determine a criterion on $a,b,c,m$ for which the weighted ternary sum $P_{a,b,c,m}:=a{p}_m(x)+b{p}_m(y)+c{p}_m(z)$ is almost universal. We also prove for some $a,b,c,m$ that the form $P_{a,b,c,m}$ is not almost universal, while it represents all possible congruence classes.