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A338323 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. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The only regular polygons that can appear are equilateral triangles, squares, and regular hexagons.

LINKS

Peter Kagey, Table of n, a(n) for n = 0..100

Code Golf Stack Exchange, Polygons in a cube

Burkard Polster, What does this prove? Some of the most gorgeous visual "shrink" proofs ever invented, Mathologer video (2020).

FORMULA

a(n) = A102698(n-1) + A334881(n) + A338322(n) for n >= 2.

EXAMPLE

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)

CROSSREFS

Cf. A102698 (equilateral triangles), A334881 (squares), A338322 (regular hexagons).

The two-dimensional case is given by A002415.

Sequence in context: A155625 A016296 A021044 * A121034 A125402 A016290

Adjacent sequences:  A338320 A338321 A338322 * A338324 A338325 A338326

KEYWORD

nonn

AUTHOR

Peter Kagey, Oct 22 2020

STATUS

approved

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Last modified June 29 18:25 EDT 2022. Contains 354913 sequences. (Running on oeis4.)