

A024406


Ordered areas of primitive Pythagorean triangles.


26



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
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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, Nonsquarefree 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



