A358883 The number of vertices in a Farey diagram of order (n,n).

5, 37, 313, 1253, 4977, 11253, 31393, 61409, 125525, 212785, 407757, 609361, 1059497, 1541005, 2328621, 3282329, 5006113, 6538721, 9545621, 12352197

Offset: 1

Author

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

Keywords

• nonn
• more

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

• 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.
• Scott R. Shannon, Image for n = 1.
• 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.
• Wikipedia, Farey sequence.

Crossrefs

Cf. A358882 (regions), A358884 (edges), A358885 (k-gons), A006842, A006843, A005728, A358887.

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.

Formula

a(n) = A358884(n) - A358882(n) + 1 by Euler's formula.