A262181 a(n) = total number of convex equilateral n-gons with corner angles of m*Pi/n (0 < m <= n).

1, 2, 1, 11, 1, 42, 64, 202, 1, 1557, 1, 5539, 32298, 30666, 1, 405200, 1, 1035642

Offset

3,2

Comments

An n-gon is a polygon with n corners and n sides, each of which is a straight line segment joining two corners. An n-gon or polygon P is said to be a simple polygon (or a Jordan polygon) if the only points of the plane belonging to two polygon edges of P are the polygon vertices 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 <= n). An n-gon is convex if it contains all the diagonal segments connecting any pair of its points. A convex polygon is sometimes strictly defined as a polygon with all its interior angles less than Pi. We use the less strict definition where every internal or interior angle is less than or equal to Pi, that is, straight angles are permitted.

Conjecture: There is only one convex equilateral n-gon for prime n.

Links

Table of n, a(n) for n=3..20.

Stuart E Anderson, C++ Program, produces postscript images of convex polygons for n, along with an unsorted list of interior angle multiples m for each polygon. One rotationally invariant representative polygon is produced in postscript.

Stuart E Anderson, for n=3, 1 solution, the equilateral triangle

Stuart E Anderson, for n=4, 2 solutions

Stuart E Anderson, for n=5, 1 solution

Stuart E Anderson, for n=6, 11 solutions

Stuart E Anderson, for n=7, 1 solution

Stuart E Anderson, for n=8, 42 solutions

Gilbert Labelle and Annie Lacasse, Closed paths whose steps are roots of unity, in FPSAC 2011, Reykjavik, Iceland DMTCS proc. AO, 2011, 599-610.

Formula

a(n) = A292355(n) for n prime or twice prime. - Andrew Howroyd, Sep 14 2017

a(n) = -(1+(-1)^n)/2 + (1/(2*n))*(A321415(n) - binomial(3*n-1, n) + Sum_{d|n} phi(n/d) * binomial(3*d-1, d)). - Andrew Howroyd, Nov 09 2018

Example

For n = 3 there is one convex n-gon, the equilateral triangle, with m angle factors (3 3 3); so a(3) = 1.
For n = 4 there are two convex n-gons, the square and a rhombus, with respective m angle factors (2 2 2 2) and (1 3 1 3); so a(4) = 2.
For n = 5, there is the regular pentagon, m factors (3 3 3 3 3); so a(5) = 1.
For n = 6 there are 11 convex n-gons; here are the m factors:(1 5 6 1 5 6), (1 6 5 1 6 5), (2 4 6 2 4 6), (2 5 5 2 5 5), (2 6 2 6 2 6), (2 6 4 2 6 4), (3 3 6 3 3 6), (3 4 5 3 4 5), (3 5 3 5 3 5), (3 5 4 3 5 4), (4 4 4 4 4 4); so a(6) = 11.

Crossrefs

A262244 for concave polygons with corner angles of m*Pi/n (0 < m < 2n), where m and n are integers.

Cf. A103314, A292355, A321415.

Sequence in context: A037916 A320390 A292355 * A309497 A281350 A252158

Adjacent sequences: A262178 A262179 A262180 * A262182 A262183 A262184

Keyword

nonn,hard,more

Author

Stuart E Anderson, Sep 14 2015

Extensions

a(10) corrected and a(12)-a(17) from Andrew Howroyd, Sep 14 2017

a(18)-a(20) from Andrew Howroyd, Nov 09 2018

Status

approved