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

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, 26531, 28424, 30405

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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 and 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. Grüner, 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).

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. A002264, A005843, A054602, A059100, A135184, A158822, A177342.

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).

Programs

Haskell

a006527 n = n * (n ^ 2 + 2) `div` 3  -- Reinhard Zumkeller, Jan 06 2014

Magma

[(n^3 + 2*n)/3: n in [0..50]]; // Vincenzo Librandi, May 15 2011

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 *)

PARI

a(n)=n*(n^2+2)/3 \\ Charles R Greathouse IV, Jul 25 2011

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)

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

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