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

%I A262181 #88 Aug 10 2025 17:29:46
%S A262181 1,2,1,11,1,42,64,202,1,1557,1,5539,32298,30666,1,405200,1,1035642
%N A262181 a(n) = total number of convex equilateral n-gons with corner angles of m*Pi/n (0 < m <= n).
%C A262181 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.
%C A262181 Conjecture: There is only one convex equilateral n-gon for prime n.
%H A262181 Stuart E Anderson, <a href="http://www.squaring.net/polygons/cvx_3-1_1_1_.pdf">for n=3, 1 solution, the equilateral triangle</a>
%H A262181 Stuart E Anderson, <a href="http://www.squaring.net/polygons/o4cvx.pdf">for n=4, 2 solutions</a>
%H A262181 Stuart E Anderson, <a href="http://www.squaring.net/polygons/cvx_5-3_3_3_3_.pdf">for n=5, 1 solution</a>
%H A262181 Stuart E Anderson, <a href="http://www.squaring.net/polygons/o6cvx.pdf">for n=6, 11 solutions</a>
%H A262181 Stuart E Anderson, <a href="http://www.squaring.net/polygons/o7cvx.pdf">for n=7, 1 solution</a>
%H A262181 Stuart E Anderson, <a href="http://www.squaring.net/polygons/o8convex.pdf">for n=8, 42 solutions</a>
%H A262181 Stuart E Anderson, <a href="/A262181/a262181.cpp.txt">n-gons.cpp</a> produces postscript images of convex polygons with n sides, or alternatively produces an unsorted list of interior angle multiples m for each polygon. One rotationally invariant representative polygon is produced in postscript and there is an option to exclude mirror image reflected polygons.
%H A262181 Gilbert Labelle and Annie Lacasse, <a href="https://doi.org/10.46298/dmtcs.2937">Closed paths whose steps are roots of unity</a>, in FPSAC 2011, Reykjavik, Iceland DMTCS proc. AO, 2011, 599-610.
%F A262181 a(n) = A292355(n) for n prime or twice prime. - _Andrew Howroyd_, Sep 14 2017
%F A262181 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
%e A262181 For n = 3 there is one convex n-gon, the equilateral triangle, with m angle factors (3 3 3); so a(3) = 1.
%e A262181 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.
%e A262181 For n = 5, there is the regular pentagon, m factors (3 3 3 3 3); so a(5) = 1.
%e A262181 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.
%Y A262181 A262244 for concave polygons with corner angles of m*Pi/n (0 < m < 2n), where m and n are integers.
%Y A262181 Cf. A103314, A292355, A321415.
%K A262181 nonn,hard,more
%O A262181 3,2
%A A262181 _Stuart E Anderson_, Sep 14 2015
%E A262181 a(10) corrected and a(12)-a(17) from _Andrew Howroyd_, Sep 14 2017
%E A262181 a(18)-a(20) from _Andrew Howroyd_, Nov 09 2018