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A339483
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Number of regular polygons that can be drawn with vertices on a centered hexagonal grid with side length n.
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1
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0, 9, 75, 294, 810, 1815, 3549, 6300, 10404, 16245, 24255, 34914, 48750, 66339, 88305, 115320, 148104, 187425, 234099, 288990, 353010, 427119, 512325, 609684, 720300, 845325, 985959, 1143450, 1319094, 1514235, 1730265, 1968624, 2230800, 2518329, 2832795
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OFFSET
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0,2
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COMMENTS
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The only regular polygons that can be drawn with vertices on the centered hexagonal grid are equilateral triangles and regular hexagons.
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LINKS
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Peter Kagey, Table of n, a(n) for n = 0..10000
Burkard Polster, What does this prove? Some of the most gorgeous visual "shrink" proofs ever invented, Mathologer video (2020).
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FORMULA
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a(n) = A000537(n) + A008893(n).
a(n) = (1/2)*(n+1)*n*(2*n+1)^2.
a(n) = 3*A180324(n).
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EXAMPLE
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There are a(2) = 75 regular polygons that can be drawn on the centered hexagonal grid with side length 2: A000537(2) = 9 regular hexagons and A008893(n) = 66 equilateral triangles.
The nine hexagons are:
* . * . * . * * .
. . . . * . . * * . * .
* . . . * . . . . . . * * . .
. . . . * . . * . . . .
* . * . * . . . .
1 1 7
which are marked with the number of ways to draw the hexagons up to translation.
The 66 equilateral triangles are:
* . . * . . * . . * . * * . . . . .
* * . . . . * . . . . . . . . . . . . . * . . *
. . . . . . * . . . . . . * . . . * . . . . . . * . . . . .
. . . . . . . . * . . . . . . . . . . . . . . .
. . . . . . . . . . . . * . . . * .
24 14 12 12 2 2
which are marked with the number of ways to draw the triangles up to translation and dihedral action of the hexagon.
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CROSSREFS
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Cf. A000537 (regular hexagons), A008893 (equilateral triangles).
Cf. A338323 (cubic grid).
Cf. A003215.
Sequence in context: A249396 A102094 A321234 * A274311 A281804 A210045
Adjacent sequences: A339480 A339481 A339482 * A339484 A339485 A339486
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KEYWORD
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nonn
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AUTHOR
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Peter Kagey, Dec 06 2020
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STATUS
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approved
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