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
Keywords
Examples
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).
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).
Programs
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Maple
# See Ionascu link for Maple program.
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Mathematica
(* See Obando link for Mathematica program. *)
Formula
a(n) approximately equals n^4.989; also lim log(a(n))/log(n) exists. - Eugen J. Ionascu, Dec 09 2006
Extensions
More terms from Hugo Pfoertner, Feb 10 2005
Edited by Ray Chandler, Sep 15 2007, Jul 27 2010
Comments