Ask HN: Weird Pattern in Pythagorean Triples – Can Anyone Explain?

Pythagorean triples are integers a, b, and c, such that a^2 + b^2 = c^2. If you look for triples such that b = a + 1, they are scarce.

But when you look at the few examples, you start to see a pattern. The value of a from a triple, divided by the value of a from the next-smaller triple, quickly converges to 5.828427.

Can anyone explain why? Why should it converge to any value? And why that value?