A358885 Table read by rows: T(n,k) = the number of regions with k sides, k >= 3, in a Farey diagram of order (n,n).

4, 48, 8, 400, 104, 1568, 456, 6216, 1848, 13944, 4256, 38760, 11976, 75768, 23480, 154440, 48248, 261072, 82184, 500464, 157440, 747480, 235528, 1298584, 410088, 1890184, 595784, 2853416, 901768, 4015552, 1274392, 6127632, 1942104, 8002552, 2537240, 11683880, 3703440, 15123800, 4790040

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.

It would be nice to have a proof (or disproof) that the number of sides is always 3 or 4.

Links

Table of n, a(n) for n=1..39.

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 = 5.

Wikipedia, Farey sequence.

Formula

Sum of row n = A358882(n).

Example

The table begins:
4;
48, 8;
400, 104;
1568, 456;
6216, 1848;
13944, 4256;
38760, 11976;
75768, 23480;
154440, 48248;
261072, 82184;
500464, 157440;
747480, 235528;
1298584, 410088;
1890184, 595784;
2853416, 901768;
4015552, 1274392;
6127632, 1942104;
8002552, 2537240;
11683880, 3703440;
15123800, 4790040;
.
.

Crossrefs

Cf. A358882 (regions), A358883 (vertices), A358884 (edges), A006842, A006843, A005728, A358889.

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: A052105 A010293 A334699 * A358889 A225987 A377055

Adjacent sequences: A358882 A358883 A358884 * A358886 A358887 A358888

Keyword

nonn,tabf

Author

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

Status

approved