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%I #13 Feb 07 2020 14:16:22
%S 120,168,180,252,280,336,396,528,540,560,600,624,792,864,880,936,1040,
%T 1050,1056,1120,1176,1224,1232,1248,1350,1368,1380,1404,1456,1620,
%U 1632,1650,1656,1710,1728,1740,1760,1764,1824,1836,1860,1960,2002,2052,2080
%N Perimeters of Pythagorean triangles that can be constructed in exactly 3 different ways.
%H Ray Chandler, <a href="/A099832/b099832.txt">Table of n, a(n) for n = 1..10000</a>
%H Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html">Pythagorean Triples and Online Calculators</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PythagoreanTriple.html">Pythagorean Triple.</a>
%H <a href="/index/Ps#PyTrip">Index entries related to Pythagorean Triples.</a>
%e a(1)=120 = 20+48+52 = 24+45+51 = 30+40+50; a(2)=168 = 21+72+75 = 24+70+74 = 42+56+70.
%t SetSystemOptions["ReduceOptions" -> {"DiscreteSolutionBound" -> 2500}]; AllPerimeterTriples[n_Integer] /; n > 0 := Module[{result = Reduce[ Reduce[{x^2 + y^2 == z^2, z > y > x > 0, Element[{x, y, z}, Integers], x + y + z == n}, {x, y, z}]]}, If[result === False, {}, Sort[{x, y, z} /. {ToRules[result]}]]]; Select[ Range[ 2500], Length[ AllPerimeterTriples[#]] == 3 &] (* _Robert G. Wilson v_, Oct 17 2012 *)
%Y Cf. A098714, A099381, A099383, A156687.
%K nonn
%O 1,1
%A _Hugo Pfoertner_, Oct 27 2004
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