  Let ${{B}_{3}}(n)$ denote the number of partition triples of $n$ where each
partition is 3-core. With the help of generating function manipulations, we
find several infinite families of arithmetic identities and congruences for
${{B}_{3}}(n)$. Moreover, let $\omega (n)$ denote the number of representations
of a nonnegative integer $n$ in the form
$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+3y_{1}^{2}+3y_{2}^{2}+3y_{3}^{2}$ with
${{x}_{1}},{{x}_{2}},{{x}_{3}},{{y}_{1}},{{y}_{2}},{{y}_{3}}\in \mathbb{Z}.$ We
find three arithmetic relations between ${{B}_{3}}(n)$ and $\omega (n)$, such
as $\omega (6n+5)=4{{B}_{3}}(6n+4).$
