  A number of the form $x(x+1)/2$ where $x$ is an integer is called a
triangular number. Suppose, $N(a_1,\cdots,a_k;n)$ and $T(a_1,\cdots,a_k;n)$
denote the number of ways $n$ can be expressed as $\sum_{i=1}^k a_ix_i^2$ and
$\sum_{i=1}^k a_i\frac{x_i(x_i+1)}{2}$, respectively. Z.-H. Sun, in \cite{4},
conjectured some relations between $T(a,b,c;n)$ and $N(a,b,c;8n+a+b+c)$. In
this paper, we prove these conjectures using theta function identities.
Moreover, we add some new triplets $(a,b,c)$ satisfying these conjectures.
