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 c_{r} is the hypotenuse of the r-th triple (ordered by, say,
the smallest leg, or by hypotenuse), Then c_{r} 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