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A006527 a(n) = (n^3 + 2*n)/3.
(Formerly M3410)
47

%I M3410

%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

%N a(n) = (n^3 + 2*n)/3.

%C Number of ways to color vertices 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 The only four of these numbers which are either triangular or hexagonal are 1, 4, 24, and 4624. 24 is hexagonal, and is the basis for MacMahon's original puzzle, and the remaining three are triangular. - _Art DuPre_, Jul 30 2012

%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, 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 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="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992.

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">1031 Generating Functions and Conjectures</a>, Université du Québec à Montréal, 1992.

%H polyforms list, <a href="http://tech.dir.groups.yahoo.com/group/polyforms/message/2086">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-1) + A000292(n-3). - _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

%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. A000292, A135184, A158822.

%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

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Last modified February 18 17:08 EST 2018. Contains 299325 sequences. (Running on oeis4.)