A073120 Areas of Pythagorean (or right) triangles with integer sides of the form (2mn, m^2 - n^2, m^2 + n^2).

6, 24, 30, 60, 84, 96, 120, 180, 210, 240, 330, 336, 384, 480, 486, 504, 546, 630, 720, 840, 924, 960, 990, 1224, 1320, 1344, 1386, 1536, 1560, 1710, 1716, 1920, 1944, 2016, 2184, 2310, 2340, 2430, 2520, 2574, 2730, 2880, 3036, 3360, 3570, 3696, 3750, 3840

Offset: 1

Author

Zak Seidov, Aug 25 2002

Keywords

• easy
• nonn

Comments

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

The sequence giving the areas of all Pythagorean triangles is A009112 (sometimes called "Pythagorean numbers").

For example, the sequence does not contain 54, the area of the Pythagorean triangle with sides (9,12,15). - Robert Israel, Apr 03 2015

See also Theorem 2 of Mohanty and Mohanty. - T. D. Noe, Sep 24 2013

Examples

			6 = 3*4/2 is the area of the right triangle with sides 3 and 4.
84 = 7*24/2 is the area of the right triangle with sides 7 and 24.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Supriya Mohanty and S. P. Mohanty, Pythagorean Numbers, Fibonacci Quarterly 28 (1990), 31-42.
• Eric Weisstein's World of Mathematics, Pythagorean Triple.
• Konstantine Hermes Zelator, A little noticed right triangle, arXiv:0804.1340 [math.GM], 2008.

Crossrefs

Cf. A009112, A002144, A003273, A046081, A057102, A024365.

Programs

Mathematica

nn = 16; t = Union[Flatten[Table[m*n*(m^2 - n^2), {m, 2, nn}, {n, m - 1}]]]; Select[t, # < nn*(nn^2 - 1) &]

Formula

a(n) = A057102(n) / 4. - Max Alekseyev, Nov 14 2008

Extensions

Description corrected by James R. Buddenhagen, Aug 10 2008, and by Max Alekseyev, Nov 12 2008

Edited by N. J. A. Sloane, Apr 06 2015