24317

Using Excel, I have found that there are six such triples with the smallest number below 100,000. I did this by calculating sqrt(n^2 + (n+1)^2) and looking for integers (using integer n's). The triples are:

3,4,5; 20,21,29; 119,120,169; 696,697,985; 4059,4060,5741; 23660,23661,33461

I really can't see any pattern there at all.