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
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