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%I #39 Jan 05 2023 11:06:01
%S 6,24,30,60,84,96,120,180,210,240,330,336,384,480,486,504,546,630,720,
%T 840,924,960,990,1224,1320,1344,1386,1536,1560,1710,1716,1920,1944,
%U 2016,2184,2310,2340,2430,2520,2574,2730,2880,3036,3360,3570,3696,3750,3840
%N Areas of Pythagorean (or right) triangles with integer sides of the form (2mn, m^2 - n^2, m^2 + n^2).
%C Equivalently, integers of the form m*n*(m^2 - n^2) where m,n are positive integers with m > n. - _James R. Buddenhagen_, Aug 10 2008
%C The sequence giving the areas of all Pythagorean triangles is A009112 (sometimes called "Pythagorean numbers").
%C For example, the sequence does not contain 54, the area of the Pythagorean triangle with sides (9,12,15). - _Robert Israel_, Apr 03 2015
%C See also Theorem 2 of Mohanty and Mohanty. - _T. D. Noe_, Sep 24 2013
%H T. D. Noe, <a href="/A073120/b073120.txt">Table of n, a(n) for n = 1..10000</a>
%H Supriya Mohanty and S. P. Mohanty, <a href="http://www.fq.math.ca/Scanned/28-1/mohanty.pdf">Pythagorean Numbers</a>, Fibonacci Quarterly 28 (1990), 31-42.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PythagoreanTriple.html">Pythagorean Triple.</a>
%H Konstantine Hermes Zelator, <a href="http://arXiv.org/abs/0804.1340">A little noticed right triangle</a>, arXiv:0804.1340 [math.GM], 2008.
%F a(n) = A057102(n) / 4. - _Max Alekseyev_, Nov 14 2008
%e 6 = 3*4/2 is the area of the right triangle with sides 3 and 4.
%e 84 = 7*24/2 is the area of the right triangle with sides 7 and 24.
%t nn = 16; t = Union[Flatten[Table[m*n*(m^2 - n^2), {m, 2, nn}, {n, m - 1}]]]; Select[t, # < nn*(nn^2 - 1) &]
%Y Cf. A009112, A002144, A003273, A046081, A057102, A024365.
%K easy,nonn
%O 1,1
%A _Zak Seidov_, Aug 25 2002
%E Description corrected by _James R. Buddenhagen_, Aug 10 2008, and by _Max Alekseyev_, Nov 12 2008
%E Edited by _N. J. A. Sloane_, Apr 06 2015
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