A358884 The number of edges in a Farey diagram of order (n,n).

8, 92, 816, 3276, 13040, 29452, 82128, 160656, 328212, 556040, 1065660, 1592368, 2768168, 4026972, 6083804, 8572272, 13075848, 17078512, 24932940, 32266036

Offset

1,1

Comments

See the linked references for further details.

The first diagram where not all edge points are connected is n = 3. For example a line connecting points (0,1/3) and (1/3,0) has equation 3*y - 6*x - 1 = 0, and as one of the x or y coefficients is greater than n (3 in this case) the line is not included.

Links

Table of n, a(n) for n=1..20.

Alain Daurat et al., About the frequencies of some patterns in digital planes. Application to area estimators. Computers & graphics. 33.1 (2009), 11-20.

Daniel Khoshnoudirad, Farey lines defining Farey diagrams and application to some discrete structures. Applicable Analysis and Discrete Mathematics. 9 (2015), 73-84.

Wikipedia, Farey sequence.

Formula

a(n) = A358882(n) + A358883(n) - 1 by Euler's formula.

Crossrefs

Cf. A358882 (regions), A358883 (vertices), A358885 (k-gons), A006842, A006843, A005728, A358888.

See A358298 for definition of Farey diagram Farey(m,n).

The Farey Diagrams Farey(m,n) are studied in A358298-A358307 and A358882-A358885, the Completed Farey Diagrams of order (m,n) in A358886-A358889.

Sequence in context: A331448 A215057 A222400 * A280159 A155615 A322650

Adjacent sequences: A358881 A358882 A358883 * A358885 A358886 A358887

Keyword

nonn,more

Author

Scott R. Shannon and N. J. A. Sloane, Dec 05 2022

Status

approved