This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A297327 #48 Feb 16 2025 08:33:52 %S A297327 6434041,89002225,865125625,89610625,353440516,29160156025, %T A297327 18989880481,37434450625,72399370000,444515646025,346008660625, %U A297327 2003915162500,9475360381201,166729268761,13110591519025,8007417968121,11201866562500,3095696620900,61956758281561 %N A297327 1/36 of the square of the basis of a primitive 3-simplex. %C A297327 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 A297327 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 A297327 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 A297327 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TrirectangularTetrahedron.html">Trirectangular Tetrahedron</a> %H A297327 Wikipedia, <a href="https://en.wikipedia.org/wiki/De_Gua%27s_theorem">De Gua's theorem</a> %F A297327 a(n) = (1/144)*(A031174(n)^2*A031175(n)^2 + A031173(n)^2*(A031174(n)^2 + A031175(n)^2)). %Y A297327 Cf. A295507, A023185, A031173, A031174, A031175. %K A297327 nonn %O A297327 1,1 %A A297327 _Ralf Steiner_, Dec 28 2017