

A262181


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


4



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

3,2


COMMENTS

An ngon is a polygon with n corners and n sides, each of which is a straight line segment joining two corners. An ngon 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 welldefined 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 ngon 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 ngon 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, 599610.


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*n1, n) + Sum_{dn} phi(n/d) * binomial(3*d1, d)).  Andrew Howroyd, Nov 09 2018


EXAMPLE

For n = 3 there is one convex ngon, the equilateral triangle, with m angle factors (3 3 3); so a(3) = 1.
For n = 4 there are two convex ngons, 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 ngons; 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



