A102698 Number of equilateral triangles with coordinates (x,y,z) in the set {0, 1,...,n}.

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

Offset

1,1

Comments

Inspired by Problem 25 on the 2005 AMC-12A Mathematics Competition, which asked for a(2).

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], 2007.

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. 129-140.

Rodrigo A. Obando, Mathematica program

Burkard Polster, What does this prove? Some of the most gorgeous visual "shrink" proofs ever invented, Mathologer video (2020).

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 A342353 A055346

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