A262244 Number of concave equilateral n-gons with corner angles of m*Pi/n (0 < m < 2n), where m and n are integers.

0, 0, 1, 22, 11, 1319, 25858

Offset: 3

Author

Stuart E Anderson, Sep 15 2015

Keywords

• nonn
• hard
• more

Comments

An n-gon is a polygon with n corners and n sides, each of which is a straight line segment joining two corners. A polygon P is said to be a simple polygon (or a Jordan polygon) if the only points of the plane belonging to multiple edges of P are the corners of P. Such a polygon has a well-defined interior and exterior. Simple polygons are topologically equivalent to a disk, hence zero angles are not allowed; allowable angles are m*Pi/n (where m and n are integers and 0 < m < 2n). A simple n-gon is concave iff at least one of its internal angles is greater than Pi, or equivalently m > n for at least one of the corners. The sum of the m-numbers (called angle factors) for the n-gon has to be n*(n-2). They are partitions of n*(n-2) into n parts with largest part n < k < 2n, and as the edges of a polygon form a closed path, the sum of unit vectors defined by the angle coordinates m/Pi is zero. The reason the m-numbers sum to n*(n-2) is that the sum of the interior angles of any n-gon is Pi*(n-2), and as angles are m*Pi/n, n = Pi.

Observation: when n is prime, m is odd and m != n.

Examples

			For n = 5, the a(5) = 1 solution is (1 3 3 1 7) in m angle factors.
For n = 7, the a(7) = 11 solutions in m angle factors are as follows: (1 11 5 3 5 5 5), (1 5 3 9 1 5 11), (1 5 5 1 11 1 11), (1 5 5 5 1 9 9), (1 5 5 5 3 5 11), (1 9 1 9 3 3 9), (1 9 3 5 1 11 5), (1 9 3 5 5 3 9), (3 3 5 5 3 3 13), (3 3 9 3 5 3 9), (3 5 5 5 5 3 9).
		

Links

• Stuart E Anderson, concave pentagon
• Stuart E Anderson, concave hexagons
• Stuart E Anderson, concave heptagons
• Stuart E Anderson, concave octagons
• Stuart E Anderson, jk-latest.cpp produces postscript images of convex, concave and intersecting polygons for n, alternatively there is a switch to produce an unsorted list of interior angle multiples m for each polygon. There is a switch to select if reflected polygons are counted. Polygons produced are all invariant under rotation, new version, Aug 04 2024.

Crossrefs

Cf. A262181 (convex equilateral polygons).

Extensions

a(9) corrected by Stuart E Anderson, Aug 04 2024