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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.
11
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
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;
.
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CROSSREFS
Cf. A358782 (regions), A358746 (vertices), A358783 (edges), A331451, A344938.
Sequence in context: A193010 A079082 A091683 * A092157 A220351 A220863
KEYWORD
nonn,tabf
AUTHOR
Scott R. Shannon, Dec 12 2022
STATUS
approved