Date: Mon, 6 Apr 1998 09:10:50 -0400
From: Alex Bogomolny
Dear Gerald:
The chapter 14, The Eternal Triangle, in Recreations in the Theory of Numbers by A.H.Beiler, is wholly devoted to Pythagorean triples with various additional properties. Specifically, on p 125, there appears a discussion on the triples with consecutive legs.
Assume cr is the hypotenuse of the r-th triple (ordered by, say, the smallest leg, or by hypotenuse), Then cr is given by
[(sqrt(2)+1)(2r+1) + (sqrt(2)-1)(2r+1)]/(2sqrt(2))
which explains your observation with regard to the factor (sqrt(2)+1)2.
To obtain the formula Beiler refers to a recurrence relation that can be surmised from the first few such triples and then proven by induction.
Following is the URL of the amazon.com page with the book's Description:
https://www.amazon.com/exec/obidos/ISBN=0486210960/001-7810978-3759339
Best regards,
Alexander Bogomolny
71546998