cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

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

Original entry on oeis.org

4, 56, 504, 2024, 8064, 18200, 50736, 99248, 202688, 343256, 657904, 983008, 1708672, 2485968, 3755184, 5289944, 8069736, 10539792, 15387320, 19913840
Offset: 1

Views

Author

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

Keywords

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

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.

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

Views

Author

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

Keywords

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.

Examples

			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;
.
.

Links

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.

Formula

Sum of row n = A358882(n).

A358887 Number of vertices 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

5, 37, 705, 4549, 42357, 94525, 531485, 1250681, 3440621, 5985201
Offset: 1

Views

Author

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

Keywords

Comments

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

Links

Crossrefs

Cf. A358888 (edges), A358886 (regions), A358889 (k-gons), A006842, A006843, A005728, A358882, A358883.
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) = A358888(n) - A358886(n) + 1 by Euler's formula.

A359690 Number of vertices 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

5, 13, 69, 289, 1971, 3997, 20371, 45751, 120957, 205299, 629847, 897801, 2334409, 3461459, 5517131, 8468061
Offset: 1

Views

Author

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

Keywords

Comments

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

Links

Crossrefs

Cf. A359691 (crossings), A359692 (regions), A359693 (edges), A359694 (k-gons), A005728, A331755, A359654, A358887, A358883, A006842, A006843.

Formula

a(n) = A359693(n) - A359692(n) + 1 by Euler's formula.

A358884 The number of edges in a Farey diagram of order (n,n).

Original entry on oeis.org

8, 92, 816, 3276, 13040, 29452, 82128, 160656, 328212, 556040, 1065660, 1592368, 2768168, 4026972, 6083804, 8572272, 13075848, 17078512, 24932940, 32266036
Offset: 1

Views

Author

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

Keywords

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

Crossrefs

Cf. A358882 (regions), A358883 (vertices), A358885 (k-gons), A006842, A006843, A005728, A358888.
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) = A358882(n) + A358883(n) - 1 by Euler's formula.

A359654 Number of vertices 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, 9, 77, 593, 6749, 15569, 93281, 222933, 623409, 1087393, 3453289, 5011009, 13271517
Offset: 1

Views

Author

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

Keywords

Comments

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

Links

Crossrefs

Cf. A359653 (regions), A359655 (edges), A359656 (k-gons), A005728, A358887, A358883, A355799, A358949, A006842, A006843.

Formula

a(n) = A359655(n) - A359653(n) + 1 by Euler's formula.

A359116 Mark the points of the Farey series F_n on a strip of paper and wrap it around a circle of circumference 1 so the endpoints 0 and 1 coincide; draw a chord between every pair of the Farey points; a(n) is the number of vertices in the resulting graph.

Original entry on oeis.org

1, 2, 5, 19, 208, 480, 3011, 7185, 20169, 35438, 111232, 162062, 422841, 633226, 1024370, 1576122, 3315790, 4240974, 8204951, 10654475, 15310713
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Dec 16 2022

Keywords

Comments

Let F_n denote the Farey series of order n: F_1 = [0, 1]; F_2 = [0, 1/2, 1]; F_3 = [0, 1/3, 1/2, 2/3, 1], F_4 = [0, 1/4, 1/3, 1/2, 2/3, 3/4, 1], etc. In general F_n consists of the points i/j with 1 <= j <= n, 0 <= i <= j, gcd(i,j) = 1, sorted and duplicates removed. Alternatively, F_n = [A006842(n,k)/A006843(n,k), k = 1..A005728(n)].
The number of terms in F_n is A005728(n). Since the endpoints coincide when we wrap the series around the circle, there are M = A005728(n) - 1 vertices on the circumference.
The planar graph we are studying, denoted by FR(n), is formed by drawing a chord between every pair of the M boundary points. FR stands for Farey Ring, a term suggested by the fairy rings found in nature.
FR(n) is analogous to the planar graph formed by drawing chords between every pair of vertices of a regular n-gon, and studied in A006533 and A007678. The difference is that in FR(n) the vertices are not equally spaced around the circle.
Just as in the case of the regular n-gon, when we count the regions in this graph, we may or may not include the regions that lie between the convex hull of the points and the bounding circle.
The first non-simple vertices that do not lie on the y = 0 or x = 0 axes occur for n = 7. If we let A = (sin(3*Pi/14) + cos(Pi/7))/(cos(3*Pi/14) + sin(Pi/7)), and B = (cos(2*Pi/7)+1)/sin(2*Pi/7), then the x coordinate of these vertices is x = +-(A*cos(3*Pi/14) - sin(3*Pi/14) - 1)/(B + A), and their y coordinate is y = -B*x - 1. These values are approximately x = +-0.1930964297 and y = -0.5990311320.

Links

Crossrefs

Cf. A359117 (regions), A359118 (edges), A359119 (k-gons).
See also A358883, A005728, A006533, A006842, A006843, A007569, A007678.

Formula

a(n) = A359118 - A359117 + 1 by Euler's formula.

A359691 Number of crossings 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

1, 7, 59, 275, 1949, 3971, 20333, 45705, 120899, 205233, 629761, 897707, 2334291, 3461329, 5516985, 8467899
Offset: 1

Views

Author

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

Keywords

Comments

The number of vertices along each edge is A005728(n). No formula for a(n) is known.
See A359690 for images of the graph.

Links

Crossrefs

Cf. A359690 (vertices), A359692 (regions), A359693 (edges), A359694 (k-gons), A005728, A159065, A331755, A359654, A358887, A358883, A006842, A006843.

Formula

a(n) = A359690(n) - 2*A005728(n).
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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