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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). 2
 0, 1, 15, 124, 603, 2173, 6204, 15201, 33149, 66002, 122410, 214186, 357189, 572385, 886117, 1330930, 1947746, 2787431, 3907866, 5380602, 7288597, 9729060, 12815704, 16677303, 21461500, 27340308, 34501149, 43160975, 53560487, 65967718, 80677972, 98029728 (list; graph; refs; listen; history; text; internal format)
 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) corresponding to 5 different tetrahedra. CROSSREFS Cf. A097125, A349296. Sequence in context: A038743 A181377 A264046 * A027839 A034271 A171220 Adjacent sequences: A349292 A349293 A349294 * A349296 A349297 A349298 KEYWORD nonn AUTHOR Giovanni Corbelli, Nov 13 2021 STATUS approved

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Last modified February 24 14:26 EST 2024. Contains 370305 sequences. (Running on oeis4.)