The number of representations by a ternary sum of triangular numbers

Abstract: For positive integers $a,b,c$, and an integer $n$, the number of integer solutions $(x,y,z) \in \mathbb Z^3$ of $a \frac{x(x-1)}{2} + b \frac{y(y-1)}{2} + c \frac{z(z-1)}{2} = n$ is denoted by $t(a,b,c;n)$. In this article, we prove some relations between $t(a,b,c;n)$ and the numbers of representations of integers by some ternary quadratic forms. In particular, we prove various conjectures given by Z. H. Sun in \cite{s}.