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%I M3854 #75 Aug 23 2022 06:24:02
%S 0,1,5,15,36,75,141,245,400,621,925,1331,1860,2535,3381,4425,5696,
%T 7225,9045,11191,13700,16611,19965,23805,28176,33125,38701,44955,
%U 51940,59711,68325,77841,88320,99825,112421,126175,141156,157435,175085,194181,214800,237021,260925,286595,314116,343575
%N Number of inequivalent ways to color vertices of a regular tetrahedron using <= n colors.
%C 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.
%C Equivalently, number of distinct tetrahedra that can be obtained by painting its faces using at most n colors. - _Lekraj Beedassy_, Dec 29 2007
%C Equals row sums of triangle A144680. - _Gary W. Adamson_, Sep 19 2008
%D 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.
%D Martin Gardner, New Mathematical Diversions from Scientific American. Simon and Schuster, NY, 1966, p. 246.
%D S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A006008/b006008.txt">Table of n, a(n) for n = 0..10000</a>
%H R. Gugisch, A. Kerber, R. Laue, M. Meringer and C. Ruecker, <a href="http://www.mathe2.uni-bayreuth.de/markus/markus.html">Kombinatorische Chemie, eine Herausforderung für Mathematik und Infomatik</a>, Spektrum 1/02, 64-67, 2002.
%H S. M. Losanitsch, <a href="/A000602/a000602_1.pdf">Die Isomerie-Arten bei den Homologen der Paraffin-Reihe</a>, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PolyhedronColoring.html">Polyhedron Coloring</a>.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F a(n) = (n^4 + 11*n^2)/12. (Replace all x_i's in the cycle index with n.)
%F Binomial transform of [1, 4, 6, 5, 2, 0, 0, 0, ...]. - _Gary W. Adamson_, Apr 23 2008
%F 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
%F 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
%F a(n) = binomial(n+3,4) + binomial(n,4). - _Collin Berman_, Jan 26 2016
%F a(n) = A000332(n+3) + A000332(n) = 2*A000332(n+3) - A006003(n) = 2*A000332(n) + A006003(n).
%F a(n) = A324999(3,n).
%F E.g.f.: (1/12)*exp(x)*x*(12 + 18*x + 6*x^2 + x^3). - _Stefano Spezia_, Jan 26 2020
%F Sum_{n>=1} 1/a(n) = (6 + 22*Pi^2 - 6*sqrt(11)*Pi*coth(sqrt(11)*Pi))/121. - _Amiram Eldar_, Aug 23 2022
%p A006008 := n->1/12*n^2*(n^2+11);
%p A006008:=-z*(z+1)*(z**2-z+1)/(z-1)**5; # conjectured by _Simon Plouffe_ in his 1992 dissertation
%t 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 *)
%o (Magma) [(n^4 + 11*n^2 )/12: n in [0..40]]; // _Vincenzo Librandi_, Aug 12 2011
%o (PARI) apply( {A006008(n)=(n^4+11*n^2)/12}, [0..50]) \\ _M. F. Hasler_, Jan 26 2020
%Y Cf. A006550, A060529.
%Y Cf. A144680. - _Gary W. Adamson_, Sep 19 2008
%Y Cf. A000332(n+3) (unoriented), A000332 (chiral), A006003 (achiral).
%Y Row 3 of A324999.
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_, Clint. C. Williams (Clintwill(AT)aol.com)
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