A297327 1/36 of the square of the basis of a primitive 3-simplex.

6434041, 89002225, 865125625, 89610625, 353440516, 29160156025, 18989880481, 37434450625, 72399370000, 444515646025, 346008660625, 2003915162500, 9475360381201, 166729268761, 13110591519025, 8007417968121, 11201866562500, 3095696620900, 61956758281561

Offset

1,1

Comments

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.

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.

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), ...

Links

Table of n, a(n) for n=1..19.

Eric Weisstein's World of Mathematics, Trirectangular Tetrahedron

Wikipedia, De Gua's theorem

Formula

a(n) = (1/144)*(A031174(n)^2*A031175(n)^2 + A031173(n)^2*(A031174(n)^2 + A031175(n)^2)).

Crossrefs

Cf. A295507, A023185, A031173, A031174, A031175.

Sequence in context: A307019 A234758 A295481 * A268843 A254174 A254167

Adjacent sequences: A297324 A297325 A297326 * A297328 A297329 A297330

Keyword

nonn

Author

Ralf Steiner, Dec 28 2017

Status

approved