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%I M3410 #136 Sep 01 2023 04:20:52
%S 0,1,4,11,24,45,76,119,176,249,340,451,584,741,924,1135,1376,1649,
%T 1956,2299,2680,3101,3564,4071,4624,5225,5876,6579,7336,8149,9020,
%U 9951,10944,12001,13124,14315,15576,16909,18316,19799,21360,23001,24724,26531,28424,30405
%N a(n) = (n^3 + 2*n)/3.
%C Number of ways to color vertices (or edges) of a triangle using <= n colors, allowing only rotations.
%C Also: dot_product (1,2,...,n)*(2,3,...,n,1), n >= 0. - _Clark Kimberling_
%C Start from triacid and attach amino acids according to the reaction scheme that describes the reaction between the active sites. See the hyperlink below on chemistry. - _Robert G. Wilson v_, Aug 02 2002
%C Starting with offset 1 = row sums of triangle A158822 and binomial transform of (1, 3, 4, 2, 0, 0, 0, ...). - _Gary W. Adamson_, Mar 28 2009
%C One-ninth of sum of three consecutive cubes: a(n) = ((n-1)^3 + n^3 + (n+1)^3)/9. - _Zak Seidov_, Jul 22 2013
%C For n > 2, number of different cubes, formed after splitting a cube in color C_1, by parallel planes in the colors C_2, C_3, ..., C_n in three spatial dimensions (in the order of the colors from a fixed vertex). Generally, in a large hypercube n^d is f(n,d) = C(n+d-1, d) + C(n, d) different small hypercubes. See below for my formula a(n) = f(n,3). - _Thomas Ordowski_, Jun 15 2014
%C a(n) is a square for n = 1, 2 & 24; and for no other values up to 10^7 (see M. Gardner). - _Michel Marcus_, Sep 06 2015
%C Number of unit tetrahedra contained in an n-scale tetrahedron composed of a tetrahedral-octahedral honeycomb. - _Jason Pruski_, Aug 23 2017
%D M. Gardner, New Mathematical Diversions from Scientific American. Simon and Schuster, NY, 1966, p. 246.
%D S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 483.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A006527/b006527.txt">Table of n, a(n) for n = 0..5000</a>
%H B. Babcock and A. van Tuyl, <a href="http://arxiv.org/abs/1109.5847">Revisiting the spreading and covering numbers</a>, arXiv preprint arXiv:1109.5847 [math.AC], 2011.
%H Richard A. Brualdi and Geir Dahl, <a href="https://arxiv.org/abs/1704.07752">Alternating Sign Matrices and Hypermatrices, and a Generalization of Latin Square</a>, arXiv:1704.07752 [math.CO], 2017. See p. 8.
%H Peter Esser?, Guenter Stertenbrink, <a href="/A006527/a006527.txt">Triangles with Mac Mahon's pieces</a>, digest of 14 messages in polyforms Yahoo group, Apr 14 - May 2, 2002.
%H Th. Gruner, A. Kerber, R. Laue and M. Meringer, <a href="ftp://ftp.mathe2.uni-bayreuth.de/meringer/pdf/MathCombChemSCCE.pdf">Mathematics for Combinatorial Chemistry</a>
%H T. P. Martin, <a href="http://dx.doi.org/10.1016/0370-1573(95)00083-6">Shells of atoms</a>, Phys. Reports, 273 (1996), 199-241, eq. (11).
%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 polyforms list, <a href="https://groups.yahoo.com/neo/groups/polyforms/conversations/topics/2035">Triangles with MacMahon's pieces</a>.
%H Taskcentre, <a href="http://www.blackdouglas.com.au/taskcentre/107mcma2.htm">McMahon's Triangles 2</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(0)=0, a(1)=1, a(2)=4, a(3)=11; for n > 3, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Harvey P. Dale_, Jun 13 2011
%F From _Paul Barry_, Mar 13 2003: (Start)
%F a(n) = 2*binomial(n+1, 3) + binomial(n, 1).
%F G.f.: x*(1+x^2)/(1-x)^4. (End)
%F a(n) = A000292(n) + A000292(n-2). - _Alexander Adamchuk_, May 20 2006
%F a(n) = n*A059100(n)/3. - _Lekraj Beedassy_, Feb 06 2007
%F a(n) = A054602(n)/3. - _Zerinvary Lajos_, Apr 20 2008
%F a(n) = ( n + Sum_{i=1..n} A177342(i) )/(n+1), with n > 0. - _Bruno Berselli_, May 19 2010
%F a(n) = A002264(A000578(n) + A005843(n)). - _Reinhard Zumkeller_, Jun 16 2011
%F a(n) = binomial(n+2, 3) + binomial(n, 3). - _Thomas Ordowski_, Jun 15 2014
%F a(n) = A000292(n) - A000292(-n). - _Bruno Berselli_, Sep 22 2016
%F E.g.f.: (x/3)*(3 + 3*x + x^2)*exp(x). - _G. C. Greubel_, Sep 01 2017
%F From _Robert A. Russell_, Oct 20 2020: (Start)
%F a(n) = 1*C(n,1) + 2*C(n,2) + 2*C(n,3), where the coefficient of C(n,k) is the number of oriented triangle colorings using exactly k colors.
%F a(n) = 2*A000292(n) - A000290(n) = 2*A000292(n-2) + A000290(n). (End)
%F Sum_{n>0} 1/a(n) = 3*(2*gamma + polygamma(0, 1-i*sqrt(2)) + polygamma(0, 1+i*sqrt(2))/4 = 1.45245201414472469745354677573358867... where i denotes the imaginary unit. - _Stefano Spezia_, Aug 31 2023
%p A006527:=z*(1+z**2)/(z-1)**4; # conjectured by _Simon Plouffe_ in his 1992 dissertation
%p with(combinat):seq(lcm(fibonacci(4,n),fibonacci(2,n))/3,n=0..42); # _Zerinvary Lajos_, Apr 20 2008
%t Table[ (n^3 + 2*n)/3, {n, 0, 45} ]
%t LinearRecurrence[{4,-6,4,-1},{0,1,4,11},46] (* or *) CoefficientList[ Series[(x+x^3)/(x-1)^4,{x,0,49}],x] (* _Harvey P. Dale_, Jun 13 2011 *)
%o (Magma) [(n^3 + 2*n)/3: n in [0..50]]; // _Vincenzo Librandi_, May 15 2011
%o (PARI) a(n)=n*(n^2+2)/3 \\ _Charles R Greathouse IV_, Jul 25 2011
%o (Haskell)
%o a006527 n = n * (n ^ 2 + 2) `div` 3 -- _Reinhard Zumkeller_, Jan 06 2014
%Y (1/12)*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.
%Y Column 1 of triangle A094414. Row 6 of the array in A107735.
%Y Cf. A002264, A005843, A054602, A059100, A135184, A158822, A177342.
%Y Cf. A000292 (unoriented), A000292(n-2) (chiral), A000290 (achiral) triangle colorings.
%Y Row 2 of A324999 (simplex vertices and facets) and A327083 (simplex edges and ridges).
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_
%E More terms from _Alexander Adamchuk_, May 20 2006
%E Corrected and replaced 5th formula from _Harvey P. Dale_, Jun 13 2011
%E Deleted an erroneous comment. - _N. J. A. Sloane_, Dec 10 2018
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