A359975 Number of regions 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, 30, 110, 479, 993, 3102, 6135, 12748, 20680, 43907, 62753, 118746, 168892, 246513, 348176, 571980, 725956, 1129035, 1426393, 1887096, 2387945, 3454566, 4123548, 5543837

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. A359974 (vertices), A359976 (edges), A359977 (k-gons), A005728, A359969, A359690, A358948, A358886.

Formula

a(n) = A359976(n) - A359974(n) + 1 by Euler's formula.