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,

We will consider the more general equation:
$X^2+qY^2+qY=Z^2$
Then, if we use the solutions of Pell's equation: $p^2-(q+1)s^2=1$
Solutions can be written in this ideal:
$X=(-p^2+2ps+(q-1)s^2)L+qs^2$
$Y=2s(p-s)L+qs^2$
$Z=(p^2-2ps+(q+1)s^2)L+qps$
And more:
$X=(p^2+2ps-(q-1)s^2)L-p^2-2ps-s^2$
$Y=2s(p+s)L-p^2-2ps-s^2$
$Z=(p^2+2ps+(q+1)s^2)L-p^2-(q+2)ps-(q+1)s^2$
$L$ - integer and given us.
