Weird Pattern in Pythagorean Triples – Can Anyone Explain?

Question

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?

ttctciyf · Accepted Answer

Googling your ratio 5.828427.. finds a seemingly related discussion[1] which identifies it as the square of the "silver ratio" i.e (1 + sqrt(2))^2[2,3]
The explanation at [1] looks ok to this non-mathematician, but best view it there I think!
1: https://math.stackexchange.com/questions/1651227/pythagorean...
2: (see calculator section at) https://www.google.com/search?q=%281+%2B+sqrt%282%29%29%5E+2
2: https://en.wikipedia.org/wiki/Silver_ratio