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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.

%I A346575 #72 Mar 24 2024 10:49:15
%S A346575 0,1,43,327,1792,6139,17607,43291,96142,193149,362383,638533,1075110,
%T A346575 1733023,2700217,4076133,5994310,8611819,12119139,16738861,22746004,
%U A346575 30449013,40212679,52452031,67651170,86348035,109166881,136796079,170024038,209707144,256814946,312433795
%N 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.
%C A346575 The existence of such a tetrahedron implies the following:
%C A346575 (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;
%C A346575 (2) a tetrahedron with such edge-lengths can be built.
%C A346575 Values were computed using a Visual Basic program with two different routines, manually checked for n = 2 and n = 3.
%C A346575 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.
%C A346575 Conjecture 2: there is no polynomial formula for a(n), as finite difference method fails.
%H A346575 Chai Wah Wu, <a href="/A346575/b346575.txt">Table of n, a(n) for n = 0..52</a>
%H A346575 Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A346575.py">Python program</a>.
%H A346575 Giovanni Corbelli, <a href="/A346575/a346575_2.txt">Visual Basic routine generating number of tetrahedra</a>.
%H A346575 Karl Wirth and Andre Dreiding, <a href="https://doi.org/10.4171/em/129">Edge lengths determining tetrahedrons</a>, Elemente der Mathematik, 64 (2009), 160-170.
%F A346575 Conjecture: Limit_{n->oo} a(n)/n^6 exists and is approximately 0.33817.
%e A346575 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).
%o A346575 (Visual Basic) ' See LINKS.
%o A346575 (Python) # See LINKS.
%Y A346575 Cf. A097125.
%Y A346575 Equivalent sequence for triples with respect to triangles: A006003.
%K A346575 nonn
%O A346575 0,3
%A A346575 _Giovanni Corbelli_, Jul 24 2021
%E A346575 a(21)-a(31) from _Lucas A. Brown_, Mar 13 2024