Prove that the diophantine equation $x^2 + (x+1)^2 = y^2$ has infinitely many solutions in positive integers.
Now, that's a Pythagorean Triplet. So, we have to prove that there are infinitely many solutions to it. I have found a few: $(x, x+1, y) = (0, 1, 1), (3, 4, 5), (20, 21, 29), (119, 120, 169), (696, 697, 985)$. I wrote a script to calculate solutions till ten million and there are only $9$ of them. Also, they alternate between even and odd $x$ though the $y$ is always odd. Somehow the $x$ turns out to be a little less than the previous $x$ times $6$.
I have no idea how to proceed. Please help.