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Number of regular polygons that can be drawn with vertices on a centered hexagonal grid with side length n.
2

%I #23 Apr 25 2024 15:24:04

%S 0,9,75,294,810,1815,3549,6300,10404,16245,24255,34914,48750,66339,

%T 88305,115320,148104,187425,234099,288990,353010,427119,512325,609684,

%U 720300,845325,985959,1143450,1319094,1514235,1730265,1968624,2230800,2518329,2832795

%N Number of regular polygons that can be drawn with vertices on a centered hexagonal grid with side length n.

%C The only regular polygons that can be drawn with vertices on the centered hexagonal grid are equilateral triangles and regular hexagons.

%H Peter Kagey, <a href="/A339483/b339483.txt">Table of n, a(n) for n = 0..10000</a>

%H Burkard Polster, <a href="https://youtu.be/sDfzCIWpS7Q">What does this prove? Some of the most gorgeous visual "shrink" proofs ever invented</a>, Mathologer video (2020).

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5, -10, 10, -5, 1).

%F a(n) = A000537(n) + A008893(n).

%F a(n) = (1/2)*(n+1)*n*(2*n+1)^2.

%F a(n) = 3*A180324(n).

%e 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.

%e The nine hexagons are:

%e * . * . * . * * .

%e . . . . * . . * * . * .

%e * . . . * . . . . . . * * . .

%e . . . . * . . * . . . .

%e * . * . * . . . .

%e 1 1 7

%e which are marked with the number of ways to draw the hexagons up to translation.

%e The 66 equilateral triangles are:

%e * . . * . . * . . * . * * . . . . .

%e * * . . . . * . . . . . . . . . . . . . * . . *

%e . . . . . . * . . . . . . * . . . * . . . . . . * . . . . .

%e . . . . . . . . * . . . . . . . . . . . . . . .

%e . . . . . . . . . . . . * . . . * .

%e 24 14 12 12 2 2

%e which are marked with the number of ways to draw the triangles up to translation and dihedral action of the hexagon.

%Y Cf. A000537 (regular hexagons), A008893 (equilateral triangles).

%Y Cf. A338323 (cubic grid).

%Y Cf. A003215.

%K nonn

%O 0,2

%A _Peter Kagey_, Dec 06 2020