A309318 a(n) is the number of polygons whose vertices are the (2*n+1)-th roots of unity and whose 2*n+1 sides all have distinct slopes.

1, 2, 24, 180, 2700, 74184, 2062800, 81067840, 3912595776

Offset: 1

Author

Ludovic Schwob, Jul 23 2019

Keywords

• nonn
• more

Comments

The polygons are counted as nonequivalent by reflection and rotation.

No even-sided polygons follow this rule.

This is the number of harmonious labelings on a cycle. See A329910 for the definition of harmonious labelings. - Wenjie Fang, Oct 14 2022

Examples

			For n=2, the a(2)=2 solutions for 2*2+1 = 5 sides are the regular pentagon and pentagram.
		

Links

• Giovanni Resta, Illustration of a(3) and a(4)
• Ludovic Schwob, On the enumeration of double cosets and self-inverse double cosets, arXiv:2506.04007 [math.CO], 2025. See p. 9.

Crossrefs

Cf. A001710 (number of polygons with n-1 sides), A329910.

Extensions

a(7)-a(9) from Giovanni Resta, Jul 27 2019