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A338322 a(n) is the number of regular hexagons with all six vertices (x,y,z) in the set {1,2,...,n}^3.

%I A338322 #26 Apr 21 2023 09:05:05
%S A338322 0,0,0,4,32,116,320,728,1472,2796,5056,8584,13792,21136,31168,45464,
%T A338322 64704,90036,122784,164472,216864,281584,360416,457400,574304,714644,
%U A338322 881312,1077612,1306720,1575088,1884928,2245336,2658592,3130028,3665376,4277376,4967424
%N A338322 a(n) is the number of regular hexagons with all six vertices (x,y,z) in the set {1,2,...,n}^3.
%H A338322 Peter Kagey, <a href="/A338322/b338322.txt">Table of n, a(n) for n = 0..100</a>
%H A338322 Code Golf Stack Exchange, <a href="https://codegolf.stackexchange.com/q/213988/53884">Polygons in a cube</a>
%H A338322 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 A338322 a(n) >= 4*(n-2)^3 for n >= 2.
%e A338322 The a(3) = 4 hexagons with integer coordinates in {1,2,3} have vertices:
%e A338322   (1,1,2), (1,2,3), (2,1,1), (2,3,3), (3,2,1), (3,3,2);
%e A338322   (1,1,2), (1,2,1), (2,1,3), (2,3,1), (3,2,3), (3,3,2);
%e A338322   (1,2,1), (1,3,2), (2,1,1), (2,3,3), (3,1,2), (3,2,3); and
%e A338322   (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), (3,2,1).
%e A338322 One of the a(5) = 116 hexagons has vertices:
%e A338322   (2,2,1), (1,4,2), (2,5,4), (4,4,5), (5,2,4), (4,1,2).
%Y A338322 Cf. A102698 (equilateral triangles), A334881 (squares), A338323 (regular polygons).
%K A338322 nonn
%O A338322 0,4
%A A338322 _Peter Kagey_, Oct 22 2020