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

1,2

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

Links

Giovanni Resta, Table of n, a(n) for n = 1..10000

Ron Knott, Pythagorean Triples and Online Calculators

Formula

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

Example

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

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 *)

Crossrefs

Cf. A020882, A020883, A020884, A020886.

Sequence in context: A313665 A015847 A224692 * A258151 A280320 A213365

Adjacent sequences: A020882 A020883 A020884 * A020886 A020887 A020888

Keyword

nonn

Author

Clark Kimberling

Extensions

Extended and corrected by David W. Wilson

Status

approved