A006008 Number of inequivalent ways to color vertices of a regular tetrahedron using <= n colors. (Formerly M3854)

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

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. - Gary W. Adamson, Sep 19 2008

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.

Martin 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 für Mathematik und Infomatik, Spektrum 1/02, 64-67, 2002.

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 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 with n.)

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

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5), with a(0)=0, a(1)=1, a(2)=5, a(3)=15, a(4)=36. - 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. - Robert A. Russell, Dec 03 2014

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

a(n) = A000332(n+3) + A000332(n) = 2*A000332(n+3) - A006003(n) = 2*A000332(n) + A006003(n).

a(n) = A324999(3,n).

E.g.f.: (1/12)*exp(x)*x*(12 + 18*x + 6*x^2 + x^3). - Stefano Spezia, Jan 26 2020

Sum_{n>=1} 1/a(n) = (6 + 22*Pi^2 - 6*sqrt(11)*Pi*coth(sqrt(11)*Pi))/121. - Amiram Eldar, Aug 23 2022

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
(PARI) apply( {A006008(n)=(n^4+11*n^2)/12}, [0..50]) \\ M. F. Hasler, Jan 26 2020

Crossrefs

Cf. A006550, A060529.

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

Cf. A000332(n+3) (unoriented), A000332 (chiral), A006003 (achiral).

Row 3 of A324999.

Sequence in context: A366971 A011933 A093802 * A325952 A086716 A046776

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