A024406 Ordered areas of primitive Pythagorean triangles.

6, 30, 60, 84, 180, 210, 210, 330, 504, 546, 630, 840, 924, 990, 1224, 1320, 1386, 1560, 1710, 1716, 2310, 2340, 2574, 2730, 2730, 3036, 3570, 3900, 4080, 4290, 4620, 4914, 5016, 5610, 5814, 6090, 6630, 7140, 7440, 7854, 7956, 7980, 7980, 8970, 8976, 9690

Offset: 1

Author

David W. Wilson

Keywords

• nonn
• easy

Comments

This sequence also gives Fibonacci's congruous numbers (or congrua) divided by 4 with multiplicities, not regarding leg exchange in the underlying primitive Pythagorean triangle. See A258150 and the example. - Wolfdieter Lang, Jun 14 2015

The squarefree part of an entry which is not squarefree is a primitive congruent number from A006991 belonging to a Pythagorean triangle with rational (not all integer) side lengths (and its companion obtained by exchanging the legs). See the W. Lang link. - Wolfdieter Lang, Oct 25 2016

Examples

			a(6) = a(7) = 210 corresponds to the area (in some squared length unit) of the primitive Pythagorean triangles (21, 20, 29) and (35, 12, 37). Fibonacci's congruum C = 840 = 210*4 belongs to the two triples [x, y, z] = [29, 41, 1] and [37, 47, 23], solving x^2 + C = y^2 and x^2 - C = z^2. - _Wolfdieter Lang_, Jun 14 2015
a(5) = 180 = 6^2*5 lead to the primitive congruent number A006991(1) = 5 from the primitive Pythagorean triangle [9, 40, 41] after division by 6: [3/2, 20/3, 41/6]. See the link for the other nonsquarefree a(n) numbers. - _Wolfdieter Lang_, Oct 25 2016
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000
• Ron Knott, Pythagorean Triples and Online Calculators
• Wolfdieter Lang, Non-squarefree entries, their congruent numbers and rational Pythagorean triangles
• Eric Weisstein's World of Mathematics, Congruum Problem

Crossrefs

Cf. A009112, A024365, A094182, A094183, A256418 (congrua), A258150.

Formula

a(n) = 6*A020885(n). - Lekraj Beedassy, Apr 30 2004

a(n) = A121728(n)*A121729(n)/2. - M. F. Hasler, Apr 16 2020