Ternary universal sums of generalized pentagonal numbers

For any $m\ge3$, every integer of the form $p_m(x)=\frac{(m-2)x^2-(m-4)x}2$ with $x \in \z$ is said to be a generalized $m$-gonal number. Let $a\le b\le c$ be positive integers. For every non negative integer $n$, if there are integers $x,y,z$ such that $n=ap_k(x)+bp_k(y)+cp_k(z)$, then the quadruple $(k;a,b,c)$ is said to be {\it universal}. Sun gave in \cite{s1} all possible quadruple candidates that are universal and proved some quadruples to be universal (see also \cite{gs}). He remains the following quadruples $(5,1,1,k)$ for $k=6,8,9,10$, $(5,1,2,8)$, and $(5,1,3,s)$ for $7 \le s \le 8$ as candidates and conjectured the universality of them. In this article we prove that the remaining 7 quadruples given above are, in fact, universal.