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

1,2

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

Table of n, a(n) for n=1..25.

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.

Formula

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

Crossrefs

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

Sequence in context: A030506 A062990 A018213 * A047661 A000649 A027173

Adjacent sequences: A359972 A359973 A359974 * A359976 A359977 A359978

Keyword

nonn,more

Author

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

Status

approved