Solve Diophantine equation with three variables part two I want to find all solutions of $$x^2+y^2+z^2-xy-yz-zx-x-y-z=0$$
Solutions need not to be primitive. I found several parametric family. For example
$(m^2, m^2+m , (m+1)^2)$
$(m^2, m^2+m+2 , (m+1)^2)$
$(m^2+1, m^2+m, (m+1)^2+1)$
$(m^2+1, m^2+m+4, (m+1)^2+1)$
$(m^2-m, m^2, (m+1)^2-(m+1))$
$(m^2-m, m^2+1, (m+1)^2-(m+1))$
$((m+2)^2-m, (m+2)^2, (m+3)^2-(m+1))$
$((m+2)^2-m, (m+2)^2+5, (m+3)^2-(m+1))$
$((m+2)^2+5, (m+2)^2+m+4, (m+3)^2+5)$
$((m+2)^2+5, (m+2)^2+m+12, (m+3)^2+5)$
$((m+3)^2-m+1, (m+3)^2+1, (m+4)^2-m)$
$((m+3)^2-m+1, (m+3)^2+8, (m+4)^2-m)$
$((m+4)^2+8, (m+4)^2+m+8, (m+5)^2+8)$
$((m+4)^2+8, (m+4)^2+m+18, (m+5)^2+8)$
$((m+3)^2+m+13, (m+5)^2+5, (m+4)^2+m+14)$
$((m+3)^2+m+13, (m+5)^2+16, (m+4)^2+m+14)$
I suspect I am missing something. These type of single-valued parametric solutions may exist in infinite numbers. Any help will be much appreciated.

I was wrong about the formula... it looks like this....
$$X^2+Y^2+Z^2=XY+XZ+ZY+X+Y+Z$$
$$X=s(3(k^2-kt+t^2)s-k-t)$$
$$Y=s(3(k^2-kt+t^2)s+2k-t)$$
$$Z=s(3(k^2-kt+t^2)s-k+2t)$$
