Geometry For Enjoyment and Challenge, by Rhoad, Milauskas and Whipple (publishers McDougal Littell/Houghton Mifflin), has some rules for generating Pythagorean triples (see also Mathworld.Wolfram.com) on p. 403:
Rule of Pythagoras: for n an odd number, the triple will be n, (n^2 – 1)/2, (n^2 + 1)/2;
Rule of Plato: for n an even number, the triple will be n, n^2/4 – 1, n^2/4 + 1;
Rule of Euclid: for m and n both even or both odd, the triple will be (m – n)/2, sqrt(mn), (m + n)/;
Rule of Maseres: for m and n any two integers, the triple will be m^2 – n^2, 2mn, m^2 + n^2.