A307923 -id:A307923 - cp's OEIS frontend

A330662 Triangle read by rows: T(n,k) is the number of polygons with 2n sides, of which k run through the center of a circle, on the circumference of which the 2n vertices of the polygon are arranged at equal spacing.

0, 0, 1, 1, 0, 2, 16, 24, 12, 8, 744, 960, 576, 192, 48, 56256, 69120, 39360, 13440, 2880, 384, 6385920, 7580160, 4204800, 1420800, 316800, 46080, 3840, 1018114560, 1178956800, 642539520, 216115200, 49190400, 7741440, 806400, 46080

Offset: 0

Ludovic Schwob, Dec 23 2019

Rotations and reflections are counted separately.
By "2*n-sided polygons" we mean the polygons that can be drawn by connecting 2*n equally spaced points on a circle.
T(0,0)=0 and T(0,1)=1 by convention.
The sequence is limited to even-sided polygons, since all odd-sided polygons have no side passing through the center.

			Triangle begins:
    0;
    0,   1;
    1,   0,   2;
   16,  24,  12,   8;
  744, 960, 576, 192, 48;

Ludovic Schwob, Table of n, a(n) for n = 0..494
Ludovic Schwob, Illustration of T(3,k), 0≤k≤3.

Row sums give A001710(2*n-1) (number of polygons with 2*n sides).
Cf. A000165 (diagonal).
Star polygons: A014106, A055684, A102302.
Cf. A309318.

T := (n, k) -> `if`(n<2, k, 2^(k-1)*binomial(n,k)*(2*n-k-1)!*hypergeom([k-n], [k-2*n+ 1], -2)):
seq(seq(simplify(T(n,k)), k=0..n),n=0..7); # Peter Luschny, Jan 07 2020

T(n,n) = 2^(n-1) * (n-1)! for all n >= 1.
T(n,0) = A307923(n) for all n>=1.
T(n,k) = binomial(n,k)* Sum_{i=k..n} (-1)^(i-k)*binomial(n-k,i-k)*(2n-1-i)!*2^(i-1), for n>=2 and 0<=k<=n.

A343369 Triangle read by rows: T(n,k) is the number of polygons formed by connecting the vertices of a regular 2n-gon such that the winding number around the center is k and with no side passing through the center.

Original entry on oeis.org

0, 0, 1, 6, 10, 0, 296, 391, 56, 1, 21580, 28298, 6132, 246, 0, 2317884, 3137098, 859536, 70389, 1012, 1, 349281380, 490054052, 158307216, 19756138, 711692, 4082, 0, 70651004192, 102443715659, 37521267472, 6221752657, 390266848, 6782563, 16368, 1

Offset: 1

Ludovic Schwob, Apr 12 2021

Polygons that differ by rotation or reflection are counted separately.

T(1,0)=0 by convention.

			Triangle begins:
      0;
      0,     1;
      6,    10,    0;
    296,   391,   56,   1;
  21580, 28298, 6132, 246,   0;

Andrew Howroyd, Table of n, a(n) for n = 1..55 (rows 1..10)
Ludovic Schwob, Illustration of T(4,k), k=0..3
Wikipedia, Winding number

Row sums are A307923.

Cf. A330660.

T(n)={
  local(Cache=Map());
  my(dir(p,q)=if(p=n&&qp-n, 1/'x, 1)));
  my(recurse(k,p,b) = my(hk=[k,p,b], z); if(!mapisdefined(Cache, hk, &z),
  z = if(k==0, p<>n, sum(q=1, 2*n-1, if(!bittest(b,q) && (q-p)%n, dir(p,q)*self()(k-1, q, b+(1<Andrew Howroyd, May 14 2021

T(2*n,2*n-1) = 1 and T(2*n+1,2*n) = 0 for all n>=1.

T(n,n-2) = 4^(n-1)-2*n for all n>=2.

a(22)-a(36) from Andrew Howroyd, May 14 2021

cp's OEIS Frontend

A330662 Triangle read by rows: T(n,k) is the number of polygons with 2n sides, of which k run through the center of a circle, on the circumference of which the 2n vertices of the polygon are arranged at equal spacing.

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A343369 Triangle read by rows: T(n,k) is the number of polygons formed by connecting the vertices of a regular 2n-gon such that the winding number around the center is k and with no side passing through the center.

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cp's OEIS Frontend

A330662 Triangle read by rows: T(n,k) is the number of polygons with 2*n sides, of which k run through the center of a circle, on the circumference of which the 2*n vertices of the polygon are arranged at equal spacing.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A343369 Triangle read by rows: T(n,k) is the number of polygons formed by connecting the vertices of a regular 2n-gon such that the winding number around the center is k and with no side passing through the center.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A330662 Triangle read by rows: T(n,k) is the number of polygons with 2n sides, of which k run through the center of a circle, on the circumference of which the 2n vertices of the polygon are arranged at equal spacing.