It is well known that all Pythagorean triples can be generated from two positive integers m and n. Iff n<m, m and n are relatively prime and not both odd, the three numbers a=m²-n², b=2mn, and c=m²+n² form a primitive Pythagorean triple (pPT). All pPT's can begenerated from the (3,4,5) triple (m=2,n=1). The three pPT generated by (3,4,5) are (21,20,29), (5,12,13) and (15,8,7).
INSTRUCTION: Change the value of m and n (see new values generated), to generate new triples (a,b,c). The new pPT's are sometimes called the "children" or "infants" of the original pPT.

For an indepth article on the background for this construction see: "The tree in Pythagoras' garden", from John Mack and Vic Czernezkyj; Australian Senior mathematics Journal, July 1,2010.