I mentioned the Google Group investigating the harder problem of finding a rectangular box with integer sides, integer diagonals, and integer main diagonal. (See http://groups.google.com/group/theperfectcuboid?lnk=iggc.)
But only while driving home did it occur to me that it's straightforward to produce lots of examples of my colleague's easier problem.
Start with your favorite primitive Pythagorean triple (a, b, c). (See my earlier post about Pythagorean triples: http://byoshiwara.blogspot.com/2009/12/blog-post.html.)
Then c is odd, so c = 2n + 1, and
a2 + b2 + [2n(n + 1)]2 = [2n(n + 1) + 1]2
For example, 32 + 42 + 122 = 132.