A033428 - OEIS

OFFSET

0,2

COMMENTS

The number of edges of a complete tripartite graph of order 3n, K_n,n,n. - Roberto E. Martinez II, Oct 18 2001

From Floor van Lamoen, Jul 21 2001: (Start)

Write 1,2,3,4,... in a hexagonal spiral around 0; then a(n) is the sequence found by reading the line from 0 in the direction 0,3,.... The spiral begins:

.

33--32--31--30

/ \

34 16--15--14 29

/ / \ \

35 17 5---4 13 28

/ / / \ \ \

36 18 6 0---3--12--27--48-->

/ / / / / / / /

37 19 7 1---2 11 26 47

\ \ \ / / /

38 20 8---9--10 25 46

\ \ / /

39 21--22--23--24 45

\ /

40--41--42--43--44

(End)

Number of edges of the complete bipartite graph of order 4n, K_n,3n. - Roberto E. Martinez II, Jan 07 2002

Also the number of partitions of 6n + 3 into at most 3 parts. - R. K. Guy, Oct 23, 2003

Also the number of partitions of 6n into exactly 3 parts. - Colin Barker, Mar 23 2015

Numbers n such that the imaginary quadratic field Q[sqrt(-n)] has six units. - Marc LeBrun, Apr 12 2006

The denominators of Hoehn's sequence (recalled by G. L. Honaker, Jr.) and the numerators of that sequence reversed. The sequence is 1/3, (1+3)/(5+7), (1+3+5)/(7+9+11), (1+3+5+7)/(9+11+13+15), ...; reduced to 1/3, 4/12, 9/27, 16/48, ... . For the reversal, the reduction is 3/1, 12/4, 27/9, 48/16, ... . - Enoch Haga, Oct 05 2007

Right edge of tables in A200737 and A200741: A200737(n, A000292(n)) = A200741(n, A100440(n)) = a(n). - Reinhard Zumkeller, Nov 21 2011

The Wiener index of the crown graph G(n) (n>=3). The crown graph G(n) is the graph with vertex set {x(1), x(2), ..., x(n), y(1), y(2), ..., y(n)} and edge set {(x(i), y(j)): 1<=i, j<=n, i/=j} (= the complete bipartite graph K(n,n) with horizontal edges removed). Example: a(3)=27 because G(3) is the cycle C(6) and 6*1 + 6*2 + 3*3 = 27. The Hosoya-Wiener polynomial of G(n) is n(n-1)(t+t^2)+nt^3. - Emeric Deutsch, Aug 29 2013

From Michel Lagneau, May 04 2015: (Start)

Integer area A of equilateral triangles whose side lengths are in the commutative ring Z[3^(1/4)] = {a + b*3^(1/4) + c*3^(1/2) + d*3^(3/4), a,b,c and d in Z}.

The area of an equilateral triangle of side length k is given by A = k^2*sqrt(3)/4. In the ring Z[3^(1/4)], if k = q*3^(1/4), then A = 3*q^2/4 is an integer if q is even. Example: 27 is in the sequence because the area of the triangle (6*3^(1/4), 6*3^(1/4), 6*3^(1/4)) is 27. (End)

a(n) is 2*sqrt(3) times the area of a 30-60-90 triangle with short side n. Also, 3 times the area of an n X n square. - Wesley Ivan Hurt, Apr 06 2016

Consider the hexagonal tiling of the plane. Extract any four hexagons adjacent by edge. This can be done in three ways. Fold the four hexagons so that all opposite faces occupy parallel planes. For all parallel projections of the resulting object, at least two correspond to area a(n) for side length of n of the original hexagons. - Torlach Rush, Aug 17 2022

LINKS

Nathaniel Johnston, Table of n, a(n) for n = 0..10000

Francesco Brenti and Paolo Sentinelli, Wachs permutations, Bruhat order and weak order, arXiv:2212.04932 [math.CO], 2022.

A. J. C. Cunningham, Factorisation of N and N' = (x^n -+ y^n) / (x -+ y) [when x-y=n], Messenger Math., 54 (1924), 17-21. [Incomplete annotated scanned copy]

Frank Ellermann, Illustration of binomial transforms.

Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.

Leo Tavares, Hexagonal illustration

Eric Weisstein's World of Mathematics, Crown Graph.

Eric Weisstein's World of Mathematics, Wiener Index.

Eric Weisstein's World of Mathematics, Unit.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA