This site is supported by donations to The OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A262181 a(n) = total number of convex equilateral n-gons 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 (list; graph; refs; listen; history; text; internal format)
 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 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 * A281350 A252158 A285996 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified March 20 09:35 EDT 2019. Contains 321345 sequences. (Running on oeis4.)