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

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

Author

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

Keywords

• nonn
• easy
• nice

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).

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.

Programs

Magma

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

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 *)

PARI

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

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