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

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

Offset

0,3

Comments

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

References

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).

Links

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).

Formula

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)

Maple

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

Mathematica

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 *)

Prog

(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

Crossrefs

(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

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

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

Status

approved