1,1

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

Ray Chandler and Eugen J. Ionascu, A characterization of all equilateral triangles in Z^3, Preprint, 2008.

Eugen J. Ionascu and Rodrigo A. Obando, Table of n, a(n) for n = 1..100

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

Rodrigo A. Obando, Mathematica program

a(n) approximately equals n^4.989; also lim ln(a(n))/ln(n) exists. - Eugen J. Ionascu (ionascu_eugen(AT)colstate.edu), Dec 09 2006

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).

See Ionascu link for Maple program

See Obando link for Mathematica program

Cf. a(n)=8*A103501, A103158 tetrahedra in lattice cube.

Sequence in context: A164755 A050799 A100431 this_sequence A055346 A159710 A100472

Adjacent sequences: A102695 A102696 A102697 this_sequence A102699 A102700 A102701

nonn

Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), Feb 04 2005

More terms from Hugo Pfoertner (hugo(AT)pfoertner.org), Feb 10 2005

Edited by Ray Chandler (rayjchandler(AT)sbcglobal.net), Sep 15 2007