A346575 a(n) is the number of 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.

0, 1, 43, 327, 1792, 6139, 17607, 43291, 96142, 193149, 362383, 638533, 1075110, 1733023, 2700217, 4076133, 5994310, 8611819, 12119139, 16738861, 22746004, 30449013, 40212679, 52452031, 67651170, 86348035, 109166881, 136796079, 170024038, 209707144, 256814946, 312433795

Offset: 0

Author

Giovanni Corbelli, Jul 24 2021

Keywords

• nonn

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.

Examples

			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).
		

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.

Crossrefs

Cf. A097125.

Equivalent sequence for triples with respect to triangles: A006003.

Programs

Python

# See LINKS.

Formula

Conjecture: Limit_{n->oo} a(n)/n^6 exists and is approximately 0.33817.

Extensions

a(21)-a(31) from Lucas A. Brown, Mar 13 2024