A349295 a(n) is the number of ordered 6-tuples (a_1,a_2,a_3,a_4,a_5,a_6) having all terms in {1,...,n} such that there exists a tetrahedron ABCD with those edge-lengths, taken in a particular order (see comments).

0, 1, 15, 124, 603, 2173, 6204, 15201, 33149, 66002, 122410, 214186, 357189, 572385, 886117, 1330930, 1947746, 2787431, 3907866, 5380602, 7288597, 9729060, 12815704, 16677303, 21461500, 27340308, 34501149, 43160975, 53560487, 65967718, 80677972, 98029728

Offset: 0

Author

Giovanni Corbelli, Nov 13 2021

Keywords

• nonn

Comments

Edges with length a_1,a_2,a_3 form a face, a_1 is opposite to a_4, a_2 is opposite to a_5, a_3 is opposite to a_6. If the a_i's are all different, then there are 24 6-tuples corresponding to the same tetrahedron. The tetrahedron is possible iff triangular inequalities hold for every face and the Cayley-Menger determinant is positive. It has been proved that if triangular inequalities hold for at least one face and the Cayley-Menger determinant is positive, then the triangular inequalities for the other three faces hold, too (see article by Wirth, Dreiding in links, (5) at page 165).

Conjecture: The ratio a(n)/n^6 decreases with n and tends to a limit which is 0.10292439+-0,00000024 (1.96 sigmas, 95% confidence level) evaluated for n=2^32 on 6.4*10^12 random 6-tuples.

Examples

			For n=2 the 6-tuples are
(1,1,1,1,1,1),
(1,1,1,2,2,2), (1,2,2,2,1,1), (2,1,2,1,2,1), (2,2,1,1,1,2),
(2,2,1,2,2,1), (2,1,2,2,1,2), (1,2,2,1,2,2),
(1,2,2,2,2,2), (2,1,2,2,2,2), (2,2,1,2,2,2), (2,2,2,1,2,2), (2,2,2,2,1,2), (2,2,2,2,2,1),
(2,2,2,2,2,2)
corresponding to A097125(1) + A097125(2) = 5 different tetrahedra.
		

Links

• Giovanni Corbelli, Table of n, a(n) for n = 0..254
• Giovanni Corbelli, FreeBasic program
• Karl Wirth and André S. Dreiding, Edge lengths determining tetrahedrons, Elemente der Mathematik, Volume 64, Issue 4, 2009, pp. 160-170.

Crossrefs

Cf. A097125, A349296, A346575.