

A102698


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


10



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
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OFFSET

1,1


COMMENTS

Inspired by Problem 25 on the 2005 AMC12A Mathematics Competition, which asked for a(2).


LINKS



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



EXTENSIONS



STATUS

approved



