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A006527
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a(n) = (n^3 + 2*n)/3.
(Formerly M3410)
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55
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0, 1, 4, 11, 24, 45, 76, 119, 176, 249, 340, 451, 584, 741, 924, 1135, 1376, 1649, 1956, 2299, 2680, 3101, 3564, 4071, 4624, 5225, 5876, 6579, 7336, 8149, 9020, 9951, 10944, 12001, 13124, 14315, 15576, 16909, 18316, 19799, 21360, 23001, 24724
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OFFSET
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0,3
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COMMENTS
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Number of ways to color vertices (or edges) of a triangle using <= n colors, allowing only rotations.
Also: dot_product (1,2,...,n)*(2,3,...,n,1), n >= 0. - Clark Kimberling
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
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
One-ninth of sum of three consecutive cubes: a(n) = ((n-1)^3 + n^3 + (n+1)^3)/9. - Zak Seidov, Jul 22 2013
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
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
Number of unit tetrahedra contained in an n-scale tetrahedron composed of a tetrahedral-octahedral honeycomb. - Jason Pruski, Aug 23 2017
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REFERENCES
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M. Gardner, New Mathematical Diversions from Scientific American. Simon and Schuster, NY, 1966, p. 246.
S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 483.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..5000
B. Babcock and A. van Tuyl, Revisiting the spreading and covering numbers, arXiv preprint arXiv:1109.5847 [math.AC], 2011.
Richard A. Brualdi, Geir Dahl, Alternating Sign Matrices and Hypermatrices, and a Generalization of Latin Square, arXiv:1704.07752 [math.CO], 2017. See p. 8.
Peter Esser?, Guenter Stertenbrink, Triangles with Mac Mahon's pieces, digest of 14 messages in polyforms Yahoo group, Apr 14 - May 2, 2002.
Th. Gruner, A. Kerber, R. Laue and M. Meringer, Mathematics for Combinatorial Chemistry
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
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
polyforms list, Triangles with MacMahon's pieces.
Taskcentre, McMahon's Triangles 2
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
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FORMULA
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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
From Paul Barry, Mar 13 2003: (Start)
a(n) = 2*binomial(n+1, 3) + binomial(n, 1).
G.f.: x*(1+x^2)/(1-x)^4. (End)
a(n) = A000292(n) + A000292(n-2). - Alexander Adamchuk, May 20 2006
a(n) = n*A059100(n)/3. - Lekraj Beedassy, Feb 06 2007
a(n) = A054602(n)/3. - Zerinvary Lajos, Apr 20 2008
a(n) = ( n + Sum_{i=1..n} A177342(i) )/(n+1), with n > 0. - Bruno Berselli, May 19 2010
a(n) = A002264(A000578(n) + A005843(n)). - Reinhard Zumkeller, Jun 16 2011
a(n) = binomial(n+2, 3) + binomial(n, 3). - Thomas Ordowski, Jun 15 2014
a(n) = A000292(n) - A000292(-n). - Bruno Berselli, Sep 22 2016
E.g.f.: (x/3)*(3 + 3*x + x^2)*exp(x). - G. C. Greubel, Sep 01 2017
From Robert A. Russell, Oct 20 2020: (Start)
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.
a(n) = 2*A000292(n) - A000290(n) = 2*A000292(n-2) + A000290(n). (End)
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MAPLE
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A006527:=z*(1+z**2)/(z-1)**4; # conjectured by Simon Plouffe in his 1992 dissertation
with(combinat):seq(lcm(fibonacci(4, n), fibonacci(2, n))/3, n=0..42); # Zerinvary Lajos, Apr 20 2008
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MATHEMATICA
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Table[ (n^3 + 2*n)/3, {n, 0, 45} ]
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 *)
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PROG
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(Magma) [(n^3 + 2*n)/3: n in [0..50]]; // Vincenzo Librandi, May 15 2011
(PARI) a(n)=n*(n^2+2)/3 \\ Charles R Greathouse IV, Jul 25 2011
(Haskell)
a006527 n = n * (n ^ 2 + 2) `div` 3 -- Reinhard Zumkeller, Jan 06 2014
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CROSSREFS
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(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.
Column 1 of triangle A094414. Row 6 of the array in A107735.
Cf. A135184, A158822.
Cf. A000292 (unoriented), A000292(n-2) (chiral), A000290 (achiral) triangle colorings.
Row 2 of A324999 (simplex vertices and facets) and A327083 (simplex edges and ridges).
Sequence in context: A099074 A014818 A328684 * A167875 A057304 A001752
Adjacent sequences: A006524 A006525 A006526 * A006528 A006529 A006530
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from Alexander Adamchuk, May 20 2006
Corrected and replaced 5th formula from Harvey P. Dale, Jun 13 2011
Deleted an erroneous comment. - N. J. A. Sloane, Dec 10 2018
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STATUS
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approved
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