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%I #19 Nov 03 2020 02:45:18
%S 0,0,14,138,640,2190,6042,13824,28400,53484,94126,156462,248568,
%T 380802,564242,813528,1146472,1581936,2143878,2857194,3749240,4854942,
%U 6210442,7856340,9832056,12194784,15002678,18312486,22183672,26693382,31909362,37916916,44802728
%N 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.
%C The only regular polygons that can appear are equilateral triangles, squares, and regular hexagons.
%H Peter Kagey, <a href="/A338323/b338323.txt">Table of n, a(n) for n = 0..100</a>
%H Code Golf Stack Exchange, <a href="https://codegolf.stackexchange.com/q/213988/53884">Polygons in a cube</a>
%H Burkard Polster, <a href="https://youtu.be/sDfzCIWpS7Q?t=799">What does this prove? Some of the most gorgeous visual "shrink" proofs ever invented</a>, Mathologer video (2020).
%F a(n) = A102698(n-1) + A334881(n) + A338322(n) for n >= 2.
%e 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.
%e An example of each shape, listed by the coordinates of their vertices:
%e Triangle: (1,2,1), (2,1,3), (3,3,2)
%e Square: (1,1,1), (2,1,1), (2,2,1), (1,2,1)
%e Hexagon: (1,1,2), (1,2,3), (2,1,1), (2,3,3), (3,2,1), (3,3,2)
%Y Cf. A102698 (equilateral triangles), A334881 (squares), A338322 (regular hexagons).
%Y The two-dimensional case is given by A002415.
%K nonn
%O 0,3
%A _Peter Kagey_, Oct 22 2020
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