OFFSET
0,3
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.
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.
EXAMPLE
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)
CROSSREFS
KEYWORD
nonn
AUTHOR
Giovanni Corbelli, Nov 13 2021
STATUS
approved