A358882 The number of regions in a Farey diagram of order (n,n).

4, 56, 504, 2024, 8064, 18200, 50736, 99248, 202688, 343256, 657904, 983008, 1708672, 2485968, 3755184, 5289944, 8069736, 10539792, 15387320, 19913840

Offset: 1

Author

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

Keywords

• nonn
• more

Comments

See A358298 and also 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, M. Tajine, M. Zouaoui, 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.
• Scott R. Shannon, Image for n = 6.
• Scott R. Shannon, Image for n = 7.
• Scott R. Shannon, Image for n = 8.
• Wikipedia, Farey sequence.

Crossrefs

Cf. A358883 (vertices), A358884 (edges), A358885 (k-gons), A006842, A006843, A005728, A358886.

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