A321415 -id:A321415 - cp's OEIS frontend

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

Stuart E Anderson, Sep 14 2015

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.

			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.

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

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

Cf. A103314, A292355, A321415.

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

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

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

A321414 Array read by antidiagonals: T(n,k) is the number of n element multisets of the 2k-th roots of unity with zero sum.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 0, 3, 0, 1, 0, 4, 2, 3, 0, 0, 5, 0, 6, 0, 1, 0, 6, 0, 10, 6, 4, 0, 0, 7, 4, 15, 0, 12, 0, 1, 0, 8, 0, 21, 2, 20, 12, 5, 0, 0, 9, 0, 28, 24, 35, 0, 21, 0, 1, 0, 10, 6, 36, 0, 64, 10, 35, 22, 6, 0, 0, 11, 0, 45, 0, 84, 84, 70, 0, 33, 0, 1

Offset: 1

Andrew Howroyd, Nov 08 2018

Equivalently, the number of closed convex paths of length n whose steps are the 2k-th roots of unity up to translation. For even n, there will be k paths of zero area consisting of n/2 steps in one direction followed by n/2 steps in the opposite direction.

			Array begins:
  =========================================================
  n\k| 1  2  3  4   5   6   7    8    9   10   11    12
  ---|-----------------------------------------------------
   1 | 0  0  0  0   0   0   0    0    0    0    0     0 ...
   2 | 1  2  3  4   5   6   7    8    9   10   11    12 ...
   3 | 0  0  2  0   0   4   0    0    6    0    0     8 ...
   4 | 1  3  6 10  15  21  28   36   45   55   66    78 ...
   5 | 0  0  6  0   2  24   0    0   54    4    0    96 ...
   6 | 1  4 12 20  35  64  84  120  183  220  286   396 ...
   7 | 0  0 12  0  10  84   2    0  270   40    0   624 ...
   8 | 1  5 21 35  70 174 210  330  657  715 1001  1749 ...
   9 | 0  0 22  0  30 236  14    0 1028  220    0  3000 ...
  10 | 1  6 33 56 128 420 462  792 2097 2010 3003  6864 ...
  11 | 0  0 36  0  70 576  56    0 3312  880    2 11976 ...
  12 | 1  7 50 84 220 926 924 1716 6039 5085 8008 24216 ...
  ...
T(5, 3) = 6 because there are 6 rotations of the following figure:
       o---o
      /     \
     o---o---o
.
T(6, 3) = 12 because there are 4 basic shapes illustrated below which with rotations and reflections give 3 + 2 + 1 + 6 = 12 convex paths.
                        o        o---o     o---o
                       / \      /     \     \   \
    o===o===o===o     o   o    o       o     o   o
                     /     \    \     /       \   \
                    o---o---o    o---o         o---o

Andrew Howroyd, Table of n, a(n) for n = 1..465

Main diagonal is A321415.
Columns include A053090(n+3), A321416, A321417, A321419.
Cf. A103306, A103314, A262181, A292355.

\\ only supports k with at most one odd prime factor.
T(n, k)={my(r=valuation(k, 2), p); polcoef(if(k>>r == 1, 1/(1-x^2)^k + O(x*x^n), if(isprimepower(k>>r, &p), ((2/(1 - x^p) - 1)/(1 - x^2 + O(x*x^n))^p)^(k/p), error("Cannot handle k=", k) )), n)}

G.f. of column k = 2^r: 1/(1 - x^2)^k - 1.

G.f. of column k = 2^r*p^e: ((2/(1 - x^p) - 1)/(1 - x^2)^p)^(k/p) - 1 for odd prime p.

cp's OEIS Frontend

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

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