Revision History for A349295
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| A349295
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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).
(history;
published version)
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#12 by N. J. A. Sloane at Fri Apr 05 00:39:01 EDT 2024
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#11 by Andrey Zabolotskiy at Thu Apr 04 21:01:46 EDT 2024
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#10 by Andrey Zabolotskiy at Thu Apr 04 21:01:37 EDT 2024
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| LINKS
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Karl Wirth and André S. Dreiding, <a href="https://emsdoi.pressorg/content10.4171/serial-article-filesem/3242129">Edge lengths determining tetrahedrons</a>, Elemente der Mathematik, Volume 64, Issue 4, 2009, pp. 160-170.
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| EXAMPLE
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corresponding to A097125(1) + A097125(2) = 5 different tetrahedra.
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| CROSSREFS
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Cf. A097125, A349296, A346575.
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| STATUS
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approved
editing
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#9 by N. J. A. Sloane at Sat Dec 25 02:42:44 EST 2021
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#8 by Jon E. Schoenfield at Sun Nov 14 13:30:31 EST 2021
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#7 by Jon E. Schoenfield at Sun Nov 14 13:30:27 EST 2021
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| NAME
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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)).
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| CROSSREFS
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Cf . A097125, A349296.
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| STATUS
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proposed
editing
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#6 by Michel Marcus at Sun Nov 14 00:55:08 EST 2021
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#5 by Michel Marcus at Sun Nov 14 00:54:24 EST 2021
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| LINKS
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Wirth, Dreiding<a href="https://ems.press/content/serial-article-files/3242">Edge lengths determining tetrahedrons</a>
Karl Wirth and André S. Dreiding, <a href="https://ems.press/content/serial-article-files/3242">Edge lengths determining tetrahedrons</a>, Elemente der Mathematik, Volume 64, Issue 4, 2009, pp. 160-170.
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| EXAMPLE
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corresponding to 5 different tetrahedra.
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| CROSSREFS
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Cf A097125, A349296.
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| STATUS
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proposed
editing
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Discussion
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| Sun Nov 14
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| Michel Marcus: please next time do NOT add a b-file before sequence is approved
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#4 by Giovanni Corbelli at Sat Nov 13 18:02:19 EST 2021
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#3 by Giovanni Corbelli at Sat Nov 13 18:00:47 EST 2021
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| NAME
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allocateda(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 forparticular Giovanniorder* (see Corbellicomments)
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| DATA
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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
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| OFFSET
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0,3
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| COMMENTS
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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.
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Giovanni Corbelli, <a href="/A349295/b349295.txt">Table of n, a(n) for n = 0..254</a>
Wirth, Dreiding<a href="https://ems.press/content/serial-article-files/3242">Edge lengths determining tetrahedrons</a>
Giovanni Corbelli, <a href="/A349295/a349295.bas.txt">FreeBasic program</a>
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| EXAMPLE
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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
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| CROSSREFS
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Cf A097125, A349296
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| KEYWORD
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allocated
nonn
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| AUTHOR
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Giovanni Corbelli, Nov 13 2021
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| STATUS
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approved
editing
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