Let us say that polynomial $P(x)$ is "friendly" if it is an integer constant, or if it has all distinct integer roots and its derivative is friendly.
While fooling around with "friendly" polynomials I found it would be useful to characterize all integer solutions to
$$ m_1^2 + m_2^2 + m_3^2 - m_1 m_2 - m_2 m_3 - m_3 m_1 = n^2 $$
There are people out there who are better in number theory problems of this sort than I am, so I am asking for help.
What I have tried: I have tried the equation in various small-integer moduli, to see if one of them would provide a constraint on the forms of the $m_i$ and $n$. But I got no real progress there.