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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. 2
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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.

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

FORMULA

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

EXAMPLE

Triangle begins:

     1;

     0,    1;

     5,    6,    1;

   140,  183,   36,   1;

  7479, 9982, 2536, 162,  1;

PROG

(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<<q)) )));

  mapput(Cache, hk, z)); z);

  my(p=recurse(2*n, 0, 0));

  if(n==0, [1], vector(n+1, i, polcoef(p, i-1)/if(i==1, 2, 1)))

}

{ for(n=0, 6, print(T(n))) } \\ Andrew Howroyd, May 16 2021

CROSSREFS

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

Cf. A343369.

Sequence in context: A113106 A171273 A157832 * A200486 A182496 A195718

Adjacent sequences:  A330657 A330658 A330659 * A330661 A330662 A330663

KEYWORD

nonn,tabl

AUTHOR

Ludovic Schwob, Dec 23 2019

EXTENSIONS

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

STATUS

approved

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Last modified October 22 22:35 EDT 2021. Contains 348180 sequences. (Running on oeis4.)