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Search: id:A006008
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| A006008 |
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Number of inequivalent ways to color vertices of a tetrahedron using <= n colors.
(Formerly M3854)
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+0
3
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| 0, 1, 5, 15, 36, 75, 141, 245, 400, 621, 925, 1331, 1860, 2535, 3381, 4425, 5696, 7225, 9045, 11191, 13700, 16611, 19965, 23805, 28176, 33125, 38701, 44955, 51940, 59711, 68325, 77841, 88320, 99825, 112421, 126175
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Here "inequivalent" refers to the rotation group of the tetrahedron, of order 12, with cycle index (x1^4 + 8*x1*x3 + 3*x2^2)/12, which is also the alternating group A_4.
Equivalently, number of distinct tetrahedra that can be obtained by painting its faces using n different colors. - Lekraj Beedassy (blekraj(AT)yahoo.com), Dec 29 2007
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REFERENCES
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S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures}, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
M. Gardner, New Mathematical Diversions from Scientific American. Simon and Schuster, NY, 1966, p. 246.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
J.-P.Delahaye, 'Le miraculeux "lemme de Burnside"', 'Le coloriage du tetraedre' pp 147 in 'Pour la Science' (French edition of 'Scientific American') No.350 December 2006 Paris.
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LINKS
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S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures}, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
R. Gugisch, A. Kerber, R. Laue, M. Meringer and C. Ruecker, Kombinatorische Chemie, eine Herausforderung fuer Mathematik und Infomatik, Spektrum 1/02, 64-67, 2002.
Eric Weisstein's World of Mathematics, Polyhedron Coloring
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FORMULA
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a(n) = (n^4 + 11n^2 )/12. (Replace all x_i's in the cycle index by n.)
Equals binomial transform of [1, 4, 6, 5, 2, 0, 0, 0,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 23 2008
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MAPLE
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A006008 := n->1/12*n^2*(n^2+11);
A006008:=-z*(z+1)*(z**2-z+1)/(z-1)**5; [Conjectured by S. Plouffe in his 1992 dissertation.]
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CROSSREFS
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Cf. A006550, A060529.
Sequence in context: A105720 A011933 A093802 this_sequence A086716 A046776 A053808
Adjacent sequences: A006005 A006006 A006007 this_sequence A006009 A006010 A006011
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KEYWORD
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nonn,easy,nice
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AUTHOR
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njas, Clint. C. Williams [ Clintwill(AT)aol.com ]
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