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A338323
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a(n) is the number of regular k-gons in three dimensions with all k vertices (x,y,z) in the set {1,2,...,n}^3.
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4
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0, 0, 14, 138, 640, 2190, 6042, 13824, 28400, 53484, 94126, 156462, 248568, 380802, 564242, 813528, 1146472, 1581936, 2143878, 2857194, 3749240, 4854942, 6210442, 7856340, 9832056, 12194784, 15002678, 18312486, 22183672, 26693382, 31909362, 37916916, 44802728
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OFFSET
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0,3
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COMMENTS
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The only regular polygons that can appear are equilateral triangles, squares, and regular hexagons.
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LINKS
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FORMULA
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EXAMPLE
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For the 3 X 3 X 3 grid, the a(3) = 138 regular polygons are A102698(3-1) = 80 triangles, A334881(3) = 54 squares, and A338322(3) = 4 hexagons.
An example of each shape, listed by the coordinates of their vertices:
Triangle: (1,2,1), (2,1,3), (3,3,2)
Square: (1,1,1), (2,1,1), (2,2,1), (1,2,1)
Hexagon: (1,1,2), (1,2,3), (2,1,1), (2,3,3), (3,2,1), (3,3,2)
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CROSSREFS
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The two-dimensional case is given by A002415.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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