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

1,1

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

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

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.

Formula

a(n) = A358884(n) - A358883(n) + 1 by Euler's formula.

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.

Sequence in context: A077122 A020540 A101540 * A224181 A026740 A191466

Adjacent sequences: A358879 A358880 A358881 * A358883 A358884 A358885

Keyword

nonn,more

Author

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

Status

approved