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%I #47 Jan 29 2023 20:09:00
%S 6434041,89002225,865125625,89610625,353440516,29160156025,
%T 18989880481,37434450625,72399370000,444515646025,346008660625,
%U 2003915162500,9475360381201,166729268761,13110591519025,8007417968121,11201866562500,3095696620900,61956758281561
%N 1/36 of the square of the basis of a primitive 3-simplex.
%C For every primitive trirectangular tetrahedron (0, a, b, c) with coprime integer sides, (b*c)^2 + (a*b)^2 + (c*a)^2 is divisible by 144.
%C The square of the basis is related by De Gua's theorem on the square of the main diagonal of a (different, not necessarily primitive) Euler brick (a*b/12=A031173(k), a*c/12=A031174(k), b*c/12=A031175(k)) also having integer sides and integer face diagonals including a trirectangular tetrahedron (0, a*b/12, a*c/12, b*c/12), such as a(1) = 6434041 = A023185(8) = A031173(8)^2 + A031174(8)^2 + A031175(8)^2.
%C By this process a cycle of primitive trirectangular tetrahedrons is defined, such as with indices k: (1 8), (2 6), (3 5), (4 7), (9 19), ...
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TrirectangularTetrahedron.html">Trirectangular Tetrahedron</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/De_Gua%27s_theorem">De Gua's theorem</a>
%F a(n) = (1/144)*(A031174(n)^2*A031175(n)^2 + A031173(n)^2*(A031174(n)^2 + A031175(n)^2)).
%Y Cf. A295507, A023185, A031173, A031174, A031175.
%K nonn
%O 1,1
%A _Ralf Steiner_, Dec 28 2017
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