A358884 -id:A358884 - cp's OEIS frontend

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

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

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.

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.

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.

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

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).

Original entry on oeis.org

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

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

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.

			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;
.
.

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.

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.

Sum of row n = A358882(n).

A358888 Number of edges formed inside a square with edge length 1 by the straight line segments mutually connecting all vertices and points that divide the sides into segments with lengths equal to the Farey series of order n = A006842(n,k)/A006843(n,k), k = 1..A005728(n).

Original entry on oeis.org

8, 92, 1744, 10612, 95460, 210020, 1161404, 2708424, 7392884, 12820768

Offset: 1

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

The number of points along each edge is given by A005728(n).

See A358886 and A358887 for images of the square.

Wikipedia, Farey sequence.

Cf. A358886 (regions), A358887 (vertices), A358889 (k-gons), A006842, A006843, A005728, A358882, A358884.

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.

a(n) = A358886(n) + A358887(n) - 1 by Euler's formula.

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

Original entry on oeis.org

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

Offset: 1

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

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.

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.

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.

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

A359693 Number of edges in a regular drawing of a complete bipartite graph where the vertex positions on each part equal the Farey series of order n.

Original entry on oeis.org

6, 24, 162, 670, 4456, 8942, 44470, 98902, 259114, 438552, 1330566, 1897164, 4893752, 7246502, 11544278, 17678880

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Jan 11 2023

The number of vertices along each edge is A005728(n). No formula for a(n) is known.

See A359690 and A359692 for images of the graph.

Eric Weisstein's World of Mathematics, Complete Bipartite Graph.
Wikipedia, Farey sequence.

Cf. A359690 (vertices), A359691 (crossings), A359692 (regions), A359694 (k-gons), A005728, A290132, A359655, A358888, A358884, A006842, A006843.

a(n) = A359690(n) + A359692(n) - 2*A005728(n) + 1 by Euler's formula.

A359655 Number of edges formed in a square with edge length 1 by straight line segments when connecting the internal edge points that divide the sides into segments with lengths equal to the Farey series of order n to the equivalent points on the opposite side of the square.

Original entry on oeis.org

4, 12, 172, 1320, 14588, 33312, 197416, 469040, 1305112, 2274592, 7172784, 10407700, 27421412

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Jan 10 2023

The number of points internal to each edge is given by A005728(n) - 2.

See A359653 and A359654 for images of the square.

Cf. A359653 (regions) A359654 (vertices), A359656 (k-gons), A005728, A358888, A358884, A355800, A358950, A006842, A006843.

a(n) = A359653(n) + A359654(n) - 1 by Euler's formula.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A358888 Number of edges formed inside a square with edge length 1 by the straight line segments mutually connecting all vertices and points that divide the sides into segments with lengths equal to the Farey series of order n = A006842(n,k)/A006843(n,k), k = 1..A005728(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A359693 Number of edges in a regular drawing of a complete bipartite graph where the vertex positions on each part equal the Farey series of order n.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A359655 Number of edges formed in a square with edge length 1 by straight line segments when connecting the internal edge points that divide the sides into segments with lengths equal to the Farey series of order n to the equivalent points on the opposite side of the square.

Views

Author

Keywords

Comments

Crossrefs

Formula