A359009 Irregular table read by rows: T(n,k) is the number of k-gons formed, k>=2, when every pair of n points, placed at the vertices of a regular n-gon, are connected by a circle and where the points lie at the ends of the circle's diameter.

1, 0, 7, 8, 4, 0, 40, 20, 6, 6, 72, 6, 0, 0, 0, 0, 0, 0, 0, 1, 0, 133, 98, 42, 7, 1, 16, 184, 56, 0, 8, 0, 342, 306, 99, 54, 0, 0, 1, 10, 510, 220, 60, 10, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 693, 858, 231, 88, 11, 11, 0, 0, 1, 24, 924, 612, 120, 60, 0, 1469, 1560, 455, 299, 13, 0, 0, 13, 0, 0, 1

Offset

2,3

Comments

Conjectures: for odd values of n all vertices are simple, other than those defining the diameters of the circles. For n > 2 and (n-2) mod 4 = 0, T(n,2) = n. For n mod 4 = 0, T(n,2) = k*n, k>=2. For odd n, T(n,2) = 0.

See A358782 for more images of the k-gons.

The author thanks Zach Shannon some of whose code was used in the generation of this sequence.

Links

Table of n, a(n) for n=2..86.

Scott R. Shannon, Image for n = 13.

Scott R. Shannon, Image for n = 21.

Formula

Sum of row n = A358782(n).

Example

The table begins:
1;
0, 7;
8, 4;
0, 40, 20, 6;
6, 72, 6, 0, 0, 0, 0, 0, 0, 0, 1;
0, 133, 98, 42, 7, 1;
16, 184, 56, 0, 8;
0, 342, 306, 99, 54, 0, 0, 1;
10, 510, 220, 60, 10, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
0, 693, 858, 231, 88, 11, 11, 0, 0, 1;
24, 924, 612, 120, 60;
0, 1469, 1560, 455, 299, 13, 0, 0, 13, 0, 0, 1;
14, 1806, 1428, 350, 98, 28, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, \\
                0, 0, 0, 1;
0, 2550, 2910, 870, 405, 75, 60, 0, 0, 0, 0, 0, 0, 1;
32, 3280, 2000, 768, 352, 0, 16;
0, 4301, 4862, 1734, 680, 102, 34, 0, 17, 0, 17, 0, 0, 0, 0, 1;
18, 4878, 4482, 1332, 324, 54, 36, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, \\
                0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
0, 6517, 7847, 2565, 1045, 190, 133, 19, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
80, 7340, 7040, 1920, 700, 0, 80;
0, 9723, 11487, 4515, 1491, 210, 168, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
.
.

Crossrefs

Cf. A358782 (regions), A358746 (vertices), A358783 (edges), A331451, A344938.

Sequence in context: A193010 A079082 A091683 * A092157 A220351 A220863

Adjacent sequences: A359006 A359007 A359008 * A359010 A359011 A359012

Keyword

nonn,tabf

Author

Scott R. Shannon, Dec 12 2022

Status

approved