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A006008 Number of inequivalent ways to color vertices of a tetrahedron using <= n colors.
(Formerly M3854)
5
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, 141156, 157435, 175085, 194181, 214800, 237021, 260925, 286595, 314116, 343575 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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 at most n colors. - Lekraj Beedassy, Dec 29 2007

Equals row sums of triangle A144680. [From Gary W. Adamson, Sep 19 2008]

Also the number of inequivalent ways to color the faces of a tetrahedron when mirror images are counted as two.

REFERENCES

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.

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.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

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.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Eric Weisstein's World of Mathematics, Polyhedron Coloring

Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

FORMULA

a(n) = (n^4 + 11*n^2 )/12. (Replace all x_i's in the cycle index by n.)

Binomial transform of [1, 4, 6, 5, 2, 0, 0, 0,...]. - Gary W. Adamson, Apr 23 2008

a(0)=0, a(1)=1, a(2)=5, a(3)=15, a(4)=36, a(n)=5*a(n-1)- 10*a(n-2)+ 10*a(n-3)-5*a(n-4)+a(n-5) [From Harvey P. Dale, Aug 11 2011]

a(n)=C(n,1)+3C(n,2)+3C(n,3)+2C(n,4)  Each term indicates the number of tetrahedra with exactly 1, 2, 3, or 4 colors.

a(n) = binomial(n+3,4) + binomial(n,4). - Collin Berman, Jan 26 2016

MAPLE

A006008 := n->1/12*n^2*(n^2+11);

A006008:=-z*(z+1)*(z**2-z+1)/(z-1)**5; [Conjectured by Simon Plouffe in his 1992 dissertation.]

MATHEMATICA

Table[(n^4+11n^2)/12, {n, 0, 40}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {0, 1, 5, 15, 36}, 40] (* Harvey P. Dale, Aug 11 2011 *)

PROG

(MAGMA) [(n^4 + 11*n^2 )/12: n in [0..40]]; // Vincenzo Librandi, Aug 12 2011

CROSSREFS

Cf. A006550, A060529.

Cf. A144680. - Gary W. Adamson, Sep 19 2008

A000332 Number when mirror images are counted as the same.

Sequence in context: A184631 A011933 A093802 * A086716 A046776 A144898

Adjacent sequences:  A006005 A006006 A006007 * A006009 A006010 A006011

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, Clint. C. Williams [ Clintwill(AT)aol.com ]

STATUS

approved

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Last modified August 20 11:34 EDT 2017. Contains 290835 sequences.