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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). 4

%I #12 Apr 05 2024 00:39:01

%S 0,1,15,124,603,2173,6204,15201,33149,66002,122410,214186,357189,

%T 572385,886117,1330930,1947746,2787431,3907866,5380602,7288597,

%U 9729060,12815704,16677303,21461500,27340308,34501149,43160975,53560487,65967718,80677972,98029728

%N 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).

%C 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).

%C 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.

%H Giovanni Corbelli, <a href="/A349295/b349295.txt">Table of n, a(n) for n = 0..254</a>

%H Giovanni Corbelli, <a href="/A349295/a349295.bas.txt">FreeBasic program</a>

%H Karl Wirth and André S. Dreiding, <a href="https://doi.org/10.4171/em/129">Edge lengths determining tetrahedrons</a>, Elemente der Mathematik, Volume 64, Issue 4, 2009, pp. 160-170.

%e For n=2 the 6-tuples are

%e (1,1,1,1,1,1),

%e (1,1,1,2,2,2), (1,2,2,2,1,1), (2,1,2,1,2,1), (2,2,1,1,1,2),

%e (2,2,1,2,2,1), (2,1,2,2,1,2), (1,2,2,1,2,2),

%e (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),

%e (2,2,2,2,2,2)

%e corresponding to A097125(1) + A097125(2) = 5 different tetrahedra.

%Y Cf. A097125, A349296, A346575.

%K nonn

%O 0,3

%A _Giovanni Corbelli_, Nov 13 2021

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