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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
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
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: A171273 A373396 A157832 * A200486 A182496 A195718
KEYWORD
nonn,tabl
AUTHOR
Ludovic Schwob, Dec 23 2019
EXTENSIONS
Terms a(21) and beyond from Andrew Howroyd, May 16 2021
STATUS
approved