OFFSET
1,6
COMMENTS
Polygons that differ by rotation or reflection are counted separately.
The polygons considered here are those that can be drawn by connecting n+2 equally spaced points on a circle (possibly self-intersecting).
The number of turns a polygon makes on itself while following its edges is called the turning number. See the Wikipedia article for additional explanation. The condition that the turns are in the same direction means that all the internal angles are less than 180 degrees (stars are allowed, but figure of eights are not).
LINKS
Ludovic Schwob, Illustration of T(6,k), 1<=k<=3
Wikipedia, Turning number
FORMULA
EXAMPLE
Triangle begins:
1;
1;
1, 1;
1, 8;
1, 29, 1;
1, 80, 47;
1, 193, 513, 1;
PROG
(PARI)
B(n, m, x)={
local(Cache=Map());
my(recurse(k, p, q, b) = my(hk=[k, p, q, b], z); if(!mapisdefined(Cache, hk, &z),
z = if(k==0, q>p && q>m, sum(j=1, n-(q-p)%n, my(r=(q+j)%n); if(!bittest(b, r), if(r<q, x, 1)*self()(k-1, q, r, b+(1<<r)) )));
mapput(Cache, hk, z)); z);
recurse(n-2, 0, m, 1+(1<<m));
}
T(n)={Vecrev(sum(i=1, n-1, B(n, i, 'x)))}
{ for(n=3, 12, print(T(n))) } \\ Andrew Howroyd, May 15 2021
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Ludovic Schwob, Apr 09 2021
EXTENSIONS
a(31)-a(49) from Andrew Howroyd, May 15 2021
STATUS
approved