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A102698
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Number of equilateral triangles with coordinates (x,y,z) in the set {0, 1,...,n}.
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3
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Inspired by Problem 25 on the 2005 AMC-12A Mathematics Competition, which asked for a(2).
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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
Rodrigo A. Obando, Mathematica program
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FORMULA
| a(n) approximately equals n^4.989; also lim ln(a(n))/ln(n) exists. - Eugen J. Ionascu (ionascu_eugen(AT)columbusstate.edu), Dec 09 2006
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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).
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MAPLE
| See Ionascu link for Maple program
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MATHEMATICA
| See Obando link for Mathematica program
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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
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KEYWORD
| nonn
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AUTHOR
| Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), Feb 04 2005
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EXTENSIONS
| More terms from Hugo Pfoertner (hugo(AT)pfoertner.org), Feb 10 2005
Edited by Ray Chandler (rayjchandler(AT)sbcglobal.net), Sep 15 2007, Jul 27 2010
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