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A024406 Ordered areas of primitive Pythagorean triangles. 26

%I

%S 6,30,60,84,180,210,210,330,504,546,630,840,924,990,1224,1320,1386,

%T 1560,1710,1716,2310,2340,2574,2730,2730,3036,3570,3900,4080,4290,

%U 4620,4914,5016,5610,5814,6090,6630,7140,7440,7854,7956,7980,7980,8970,8976,9690

%N Ordered areas of primitive Pythagorean triangles.

%C 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

%C 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

%H Giovanni Resta, <a href="/A024406/b024406.txt">Table of n, a(n) for n = 1..10000</a>

%H Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html">Pythagorean Triples and Online Calculators</a>

%H Wolfdieter Lang, <a href="/A024406/a024406.pdf">Non-squarefree entries, their congruent numbers and rational Pythagorean triangles</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CongruumProblem.html">Congruum Problem </a>

%F a(n) = 6*A020885(n). - _Lekraj Beedassy_, Apr 30 2004

%F a(n) = A121728(n)*A121729(n)/2. - _M. F. Hasler_, Apr 16 2020

%e 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

%e 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

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

%K nonn,easy

%O 1,1

%A _David W. Wilson_

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Last modified February 24 13:13 EST 2021. Contains 341569 sequences. (Running on oeis4.)