Quadratic Diophantine equation in three variables How would one determine solutions to the following quadratic Diophantine equation in three variables:
$$x^2 + n^2y^2 \pm n^2y = z^2$$
where n is a known integer and $x$, $y$, and $z$ are unknown positive integers to be solved.
Ideally there would be a parametric solution for $x$, $y$, and $z$.
[Note that the expression $y^2 + y$ must be an integer from the series {2, 6, 12, 20, 30, 42 ...} and so can be written as either $y^2 + y$ or $y^2 - y$ (e.g., 12 = $3^2 + 3$ and 12 = $4^2 - 4$).  So I have written this as +/- in the equation above.]
Thanks,

For the case when the equation: $X^2+qY^2+qY=Z^2$
factor $q$ - is not a square, then solutions can yrazit using Pell's equation: $p^2-qs^2=1$
Then there is another solution:
$X=X$
$Y=2psX-p^2$
$Z=Xp^2-qps+qXs^2$
