A359977 Irregular table read by rows: T(n,k) is the number of k-gons, k>=3, formed inside a right triangle by the straight line segments mutually connecting all vertices and points on the two shorter edges whose positions on one edge equal the Farey series of order n while on the other they divide its length into n equal segments.

1, 5, 20, 8, 2, 50, 57, 3, 169, 274, 31, 5, 303, 646, 41, 2, 1, 889, 2011, 179, 21, 2, 1685, 4025, 388, 33, 4, 3466, 8283, 925, 67, 7, 5624, 13442, 1498, 106, 9, 1, 11896, 27907, 3718, 354, 30, 2, 16976, 40100, 5182, 461, 33, 1, 32506, 73806, 11249, 1118, 61, 6, 46187, 104453, 16380, 1747, 123, 1, 1

Offset: 1

Author

Scott R. Shannon and N. J. A. Sloane, Jan 20 2023

Keywords

• nonn
• tabf

Comments

The number of vertices on the edge with point positions equaling the Farey series of order n is A005728(n). No formula for a(n) is known.

See A359974 and A359975 for images of the triangle.

This graph is related to the 'Farey fan' given in the reference.

Examples

			The table begins:
1;
5;
20, 8, 2;
50, 57, 3;
169, 274, 31, 5;
303, 646, 41, 2, 1;
889, 2011, 179, 21, 2;
1685, 4025, 388, 33, 4;
3466, 8283, 925, 67, 7;
5624, 13442, 1498, 106, 9, 1;
11896, 27907, 3718, 354, 30, 2;
16976, 40100, 5182, 461, 33, 1;
32506, 73806, 11249, 1118, 61, 6;
46187, 104453, 16380, 1747, 123, 1, 1;
67117, 152534, 24159, 2511, 181, 10, 1;
95276, 213798, 34962, 3824, 295, 21;
.
.
		

References

• McIlroy, M. D. "A Note on Discrete Representation of Lines". AT&T Technical Journal, 64 (1985), 481-490.

Crossrefs

Cf. A359974 (vertices), A359975 (regions), A359976 (edges), A005728, A359971, A359694, A358951, A358889.

Formula

Sum of row n = A359975(n).