

A102698


Number of equilateral triangles with coordinates (x,y,z) in the set {0, 1,...,n}.


5



8, 80, 368, 1264, 3448, 7792, 16176, 30696, 54216, 90104, 143576, 220328, 326680, 471232, 664648, 916344, 1241856, 1655208, 2172584, 2812664, 3598664, 4553800, 5702776, 7075264, 8705088, 10628928, 12880056, 15496616, 18523472, 22003808
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OFFSET

1,1


COMMENTS

Inspired by Problem 25 on the 2005 AMC12A Mathematics Competition, which asked for a(2).


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

Eugen J. Ionascu and Rodrigo A. Obando, Table of n, a(n) for n = 1..100
Ray Chandler and Eugen J. Ionascu, A characterization of all equilateral triangles in Z^3, arXiv:0710.0708 [math.NT].
Eugen J. Ionascu, Maple program
Eugen J. Ionascu, A parametrization of equilateral triangles having integer coordinates, J. Integer Seqs., Vol. 10 (2007), #07.6.7.
Eugen J. Ionascu, Counting all equilateral triangles in {0,1,...,n}^3, Acta Mathematica Universitatis Comenianae, Vol. LXXVII, 1 (2008) p. 129140.
Rodrigo A. Obando, Mathematica program


FORMULA

a(n) approximately equals n^4.989; also lim log(a(n))/log(n) exists.  Eugen J. Ionascu, Dec 09 2006


EXAMPLE

a(1) = 8 because in the unit cube, equilateral triangles are formed by cutting off any one of the 8 corners.
a(2) = 80 because there are 8 unit cubes with 8 each, 8 larger triangles (analogous to the 8 in the unit cube, but twice as big) and also 8 triangles of side length sqrt(6).


MAPLE

See Ionascu link for Maple program.


MATHEMATICA

See Obando link for Mathematica program.


CROSSREFS

Cf. a(n)=8*A103501, A103158 tetrahedra in lattice cube.
Sequence in context: A050799 A100431 A173116 * A190019 A055346 A159710
Adjacent sequences: A102695 A102696 A102697 * A102699 A102700 A102701


KEYWORD

nonn


AUTHOR

Joshua Zucker, Feb 04 2005


EXTENSIONS

More terms from Hugo Pfoertner, Feb 10 2005
Edited by Ray Chandler, Sep 15 2007, Jul 27 2010


STATUS

approved



