

A008893


Number of equilateral triangles formed by triples of points taken from a hexagonal chunk of side n in the hexagonal lattice.


6



0, 8, 66, 258, 710, 1590, 3108, 5516, 9108, 14220, 21230, 30558, 42666, 58058, 77280, 100920, 129608, 164016, 204858, 252890, 308910, 373758, 448316, 533508, 630300, 739700, 862758, 1000566, 1154258, 1325010, 1514040, 1722608, 1952016, 2203608, 2478770
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,2


COMMENTS

The hexagonal lattice is the familiar 2dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice. Here we consider a hexagonal chunk of the lattice in which each bounding edge contains n+1 points.


REFERENCES

Mohammad K. Azarian, A Trigonometric Characterization of Equilateral Triangle, Problem 336, Mathematics and Computer Education, Vol. 31, No. 1, Winter 1997, p. 96. Solution published in Vol. 32, No. 1, Winter 1998, pp. 8485.
Mohammad K. Azarian, Equating Distances and Altitude in an Equilateral Triangle, Problem 316, Mathematics and Computer Education, Vol. 28, No. 3, Fall 1994, p. 337. Solution published in Vol. 29, No. 3, Fall 1995, pp. 324325.


LINKS

Nathaniel Johnston, Table of n, a(n) for n = 0..10000
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
N. J. A. Sloane, Illustration for a(1)=8. [The drawing was made for a different offset, so it says a(2)=8.]


FORMULA

a(n) = n*(n+1)*(7*n^2+7*n+2)/4.
G.f.: 2*x*(4*x^2+13*x+4)/(x1)^5 [From Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009]


PROG

(Maxima) A008893(n):=n*(n+1)*(7*n^2+7*n+2)/4$
makelist(A008893(n), n, 0, 30); /* Martin Ettl, Nov 03 2012 */


CROSSREFS

Cf. A045949, A152041.
Sequence in context: A226126 A039329 A230736 * A168302 A121782 A212784
Adjacent sequences: A008890 A008891 A008892 * A008894 A008895 A008896


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane, R. K. Guy


EXTENSIONS

Edited May 29 2012 by N. J. A. Sloane, May 29 2012


STATUS

approved



