OFFSET
0,3
COMMENTS
The existence of such a tetrahedron implies the following:
(1) there exists at least one permutation (a_i1,a_i2,a_i3,a_i4,a_i5,a_i6) such that triangular inequalities hold for (a_i1,a_i2,a_i3) (BCD), (a_i1,a_i4,a_i5) (ABC), (a_i2,a_i5,a_i6) (ACD) and (a_i3,a_i6,a_i4) (ABD), where we have a_i1=BC, a_i2=CD, a_i3=DB, a_i4=AB, a_i5=AC, a_i6=AD;
(2) a tetrahedron with such edge-lengths can be built.
Values were computed using a Visual Basic program with two different routines, manually checked for n = 2 and n = 3.
Conjecture 1: a(n)/n^6 tends to a limit which is 0.338170 +- 0.000017 (confidence level 95%). This number has been evaluated with a Monte-Carlo test on 3 billion sextuples with random values in (0,1) which simulate n -> oo.
Conjecture 2: there is no polynomial formula for a(n), as finite difference method fails.
LINKS
Chai Wah Wu, Table of n, a(n) for n = 0..52
Lucas A. Brown, Python program.
Giovanni Corbelli, Visual Basic routine generating number of tetrahedra.
Karl Wirth and Andre Dreiding, Edge lengths determining tetrahedrons, Elemente der Mathematik, 64 (2009), 160-170.
FORMULA
Conjecture: Limit_{n->oo} a(n)/n^6 exists and is approximately 0.33817.
EXAMPLE
For a(2)=43 the solutions are (1,1,1,1,1,1), all 20 permutations of (1,1,1,2,2,2), all 15 permutations of (1,1,2,2,2,2), all 6 permutations of (1,2,2,2,2,2) and (2,2,2,2,2,2).
PROG
(Visual Basic) ' See LINKS.
(Python) # See LINKS.
CROSSREFS
KEYWORD
nonn
AUTHOR
Giovanni Corbelli, Jul 24 2021
EXTENSIONS
a(21)-a(31) from Lucas A. Brown, Mar 13 2024
STATUS
approved