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

1,1

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

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

Formula

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

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

Example

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

Crossrefs

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

Sequence in context: A014203 A044083 A239978 * A024365 A057229 A120734

Adjacent sequences: A024403 A024404 A024405 * A024407 A024408 A024409

Keyword

nonn,easy

Author

David W. Wilson

Status

approved