recognize the identity:
If you let $x = a - b$, and $y = a + b$, and $z = (a^2+b^2)$
we then have:
$$ x^2 + y^2 = 2z^2$$
Which connects back to your equation.
In response to your other question, there can exist 3 natural numbers for which it can form both a arithmetic and geometric sequence (not in succesion).
i.e. $2 + 4 + 6 + .....$ Arithmetic Sequence, Common Difference: 2, First term: 2
$2 + 4 + 8 + 16 + .....$ Geometric Sequence, Common Ratio: 2, First term: 2
As you can see, the first 3 terms of this Geometric sequence are also part of an arithmetic sequence (again, not in succession).