A359694 Irregular table read by rows: T(n,k) is the number of k-gons, k>=3, in a regular drawing of a complete bipartite graph where the vertex positions on each part equal the Farey series of order n.

2, 10, 2, 70, 24, 218, 160, 4, 1254, 1068, 148, 16, 2254, 2414, 252, 26, 10082, 11760, 1980, 266, 12, 21410, 25958, 5096, 648, 36, 4, 53422, 68208, 14360, 1980, 168, 20, 86986, 118922, 24028, 3056, 248, 12, 0, 2, 255678, 346676, 84344, 12774, 1132, 110, 4, 2, 365674, 493530, 119820, 18600, 1624, 112, 4

Offset: 1

Author

Scott R. Shannon and N. J. A. Sloane, Jan 11 2023

Keywords

• nonn
• tabf

Comments

The number of vertices along each edge is A005728(n). No formula is known.

See A359692 for other images of the graph.

Examples

			The table begins:
2;
10, 2;
70, 24;
218, 160, 4;
1254, 1068, 148, 16;
2254, 2414, 252, 26;
10082, 11760, 1980, 266, 12;
21410, 25958, 5096, 648, 36, 4;
53422, 68208, 14360, 1980, 168, 20;
86986, 118922, 24028, 3056, 248, 12, 0, 2;
255678, 346676, 84344, 12774, 1132, 110, 4, 2;
365674, 493530, 119820, 18600, 1624, 112, 4;
917478, 1244492, 334096, 57080, 5700, 478, 16, 4;
1335398, 1862666, 495536, 82642, 8096, 676, 24, 6;
2107042, 2989864, 788340, 128378, 12536, 932, 52, 4;
3195474, 4557430, 1230300, 205352, 20516, 1664, 80, 4;
.
.
		

Links

• Scott R. Shannon, Image for n = 7.
• Eric Weisstein's World of Mathematics, Complete Bipartite Graph.
• Wikipedia, Farey sequence.

Crossrefs

Cf. A359690 (vertices), A359691 (crossings), A359692 (regions), A359693 (edges), A005728, A290131, A359653, A358886, A358882, A006842, A006843.

Formula

Sum of row n = A359692(n).