A020885 Ordered areas (divided by 6) of primitive Pythagorean triangles (with multiple entries).

1, 5, 10, 14, 30, 35, 35, 55, 84, 91, 105, 140, 154, 165, 204, 220, 231, 260, 285, 286, 385, 390, 429, 455, 455, 506, 595, 650, 680, 715, 770, 819, 836, 935, 969, 1015, 1105, 1190, 1240, 1309, 1326, 1330, 1330, 1495, 1496, 1615, 1729, 1771, 1785, 1820, 1925

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Comments

Since squares are 0 or 1 under both mod 3 and mod 4, for the Pythagorean equation A^2 + B^2 = C^2 to hold, each of 3 and 4 divides either of leg A or leg B, so that area A*B/2 is divisible by 3*4/2 = 6. - Lekraj Beedassy, Apr 30 2004

From Wolfdieter Lang, Jun 14 2015: (Start)

This sequence gives the area/6 (in some squared length unit) of primitive Pythagorean triangles with multiplicities modulo leg exchange. See the example.

This sequence also gives Fibonacci's congruous numbers divided by 24, with multiplicities and ordered nondecreasingly. See A258150.

(End)

It appears that this sequence gives the list of dimensions of irreducible unitary representations of the Lie group SO(5). - Antoine Bourget, Mar 30 2022

Examples

			a(6) = a(7) = 35 from the two Pythagorean triangles (A,B,C) = (21, 20, 29)  and (35, 12, 37) with area 210. Triangles (20, 21, 29) and (12, 35, 37) are not counted (leg exchange). - _Wolfdieter Lang_, Jun 14 2015
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000
• Ron Knott, Pythagorean Triples and Online Calculators

Crossrefs

Cf. A020882, A020883, A020884, A020886.

Programs

Mathematica

Take[Sort[(Times@@#)/12&/@({Times@@#,(Last[#]^2-First[#]^2)/2}&/@ Select[ Subsets[Range[1,41,2],{2}],GCD@@#==1&])],60] (* Harvey P. Dale, Feb 27 2012 *)

Formula

a(n) = A024406(n)/6.

Extensions

Extended and corrected by David W. Wilson