

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
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OFFSET

0,4


COMMENTS

Polygons that differ by rotation or reflection are counted separately.
By "2*n+1sided 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 evensided 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 SchlĂ¤fli 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>=pn, 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()(k1, 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, i1)/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



