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
KEYWORD
nonn,tabl
AUTHOR
Ludovic Schwob, Dec 23 2019
EXTENSIONS
Terms a(21) and beyond from Andrew Howroyd, May 16 2021
STATUS
approved