A330660 Triangle read by rows: T(n,k) is the number of polygons formed by connecting the vertices of a regular {2*n+1}-gon such that they make k turns around the center point.

1, 0, 1, 5, 6, 1, 140, 183, 36, 1, 7479, 9982, 2536, 162, 1, 636944, 880738, 267664, 28381, 672, 1, 79661322, 113973276, 39717471, 5860934, 285078, 2718, 1, 13781863080, 20321795499, 7893750308, 1475570241, 113442968, 2712595, 10908, 1

Offset: 0

Author

Ludovic Schwob, Dec 23 2019

Keywords

• nonn
• tabl

Comments

Polygons that differ by rotation or reflection are counted separately.

By "2*n+1-sided polygons" we mean the polygons that can be drawn by connecting 2*n+1 equally spaced points on a circle.

T(0,0)=1 by convention.

T(n,k) is the number of polygons with 2*n+1 sides whose winding number around the center point is k.

Only polygons with an odd number of sides are considered, since even-sided polygons may have diagonals passing through the center point.

Examples

			Triangle begins:
     1;
     0,    1;
     5,    6,    1;
   140,  183,   36,   1;
  7479, 9982, 2536, 162,  1;
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..54
• Ludovic Schwob, Illustration of T(3,k), 0 <= k <= 3
• Dan Sunday, Inclusion of a Point in a Polygon, (2001).
• Wikipedia, Winding number

Crossrefs

Row sums give A001710(2*n) (number of polygons with 2*n+1 sides).

Cf. A343369.

Programs

PARI

T(n)={
  local(Cache=Map());
  my(dir(p, q)=if(p<=n, if(q>n&&q<=p+n, 'x, 1), if(q<=n&&q>=p-n, 1/'x, 1)));
  my(recurse(k, p, b) = my(hk=[k, p, b], z); if(!mapisdefined(Cache, hk, &z),
  z = if(k==0, 1, sum(q=1, 2*n, if(!bittest(b, q), dir(p, q)*self()(k-1, q, b+(1<Andrew Howroyd, May 16 2021

Formula

T(n,n)=1 for all n >= 0: The only solution is the polygon with Schläfli symbol {2n*1/n}.

Extensions

Terms a(21) and beyond from Andrew Howroyd, May 16 2021