A359974 Number of vertices 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.

3, 6, 26, 93, 424, 876, 2785, 5542, 11575, 18761, 40249, 57399, 109376, 155965, 227884, 322377, 532454, 676282, 1056010, 1334975, 1767798, 2240664, 3252047, 3882192, 5226897

Offset: 1

Author

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

Keywords

• nonn
• more

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.

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

References

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

Links

• Scott R. Shannon, Image for n = 2.
• Scott R. Shannon, Image for n = 3.
• Scott R. Shannon, Image for n = 4.
• Scott R. Shannon, Image for n = 5.
• Scott R. Shannon, Image for n = 6.

Crossrefs

Cf. A359975 (regions), A359976 (edges), A359977 (k-gons), A005728, A359968, A359690, A358949, A358887.

Formula

a(n) = A359976(n) - A359975(n) + 1 by Euler's formula.