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

1,1

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.

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.

Formula

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

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.

Sequence in context: A006442 A208675 A084212 * A323219 A176818 A359643

Adjacent sequences: A358880 A358881 A358882 * A358884 A358885 A358886

Keyword

nonn,more

Author

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

Status

approved