Almost universal ternary sums of polygonal numbers

For a natural number $m$, generalized $m$-gonal numbers are those numbers of the form $p_m(x)=\frac{(m-2)x^2-(m-4)x}{2}$ with $x\in \mathbb{Z}$. In this paper we establish conditions on $m$ for which the ternary sum $p_m(x)+p_m(y)+p_m(z)$ is almost universal.