Few and Far Between?

This open investigation follows on from the problem Generating Triples. It can be taken to many levels of complexity.

The well-known Pythagorean Triple ${3,4,5}$ has the property that the two smaller numbers differ by 1.
 
Can you find any more Pythagorean Triples where the two smaller numbers differ by 1? Investigate your findings.

For more investigations see our Stage 5 pages.