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 1-10 of 10 results.

A358886 Number of regions 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

4, 56, 1040, 6064, 53104, 115496, 629920, 1457744, 3952264, 6835568
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).
We call this graph the Completed Farey Diagram of order (n,k). The (ordinary) Farey diagram Farey(n,k) is a subgraph. In the latter graph, not all pairs of boundary points are joined by lines.

Links

Crossrefs

Cf. A358888 (edges), A358887 (vertices), A358889 (k-gons), A006842, A006843, A005728, A358882.
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) - A358887(n) + 1 by Euler's formula.

A358889 Table read by rows: T(n,k) = number of k-gons, k >= 3, 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,m)/A006843(n,m), m = 1..A005728(n).

Original entry on oeis.org

4, 48, 8, 712, 304, 24, 3368, 2400, 280, 16, 27424, 20360, 4784, 504, 32, 56000, 47088, 10912, 1400, 88, 8, 292424, 255608, 69368, 11504, 960, 56, 658800, 590208, 175856, 30160, 2496, 200, 24, 1748112, 1593912, 506496, 93584, 9616, 520, 24, 2981448, 2778456, 890368, 166912, 17192, 1144, 48
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).

Examples

			The table begins:
 4;
 48,      8;
 712,     304,     24;
 3368,    2400,    280,    16;
 27424,   20360,   4784,   504,    32;
 56000,   47088,   10912,  1400,   88,    8;
 292424,  255608,  69368,  11504,  960,   56;
 658800,  590208,  175856, 30160,  2496,  200,  24;
 1748112, 1593912, 506496, 93584,  9616,  520,  24;
 2981448, 2778456, 890368, 166912, 17192, 1144, 48;
.
.

Links

Crossrefs

Cf. A358886 (regions), A358887 (vertices), A358888 (edges), A006842, A006843, A005728, A358885.
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 = A358886(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

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).
See A358886 and A358887 for images of the square.

Links

Crossrefs

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.

Formula

a(n) = A358886(n) + A358887(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.

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

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), 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.

Formula

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

A358949 Number of vertices formed inside a triangle 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

3, 10, 148, 1111, 9568, 23770, 126187, 308401, 855145, 1521733, 4591405, 6831040
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Dec 07 2022

Keywords

Comments

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

Links

Crossrefs

Cf. A358948 (regions), A358950 (edges), A358951 (k-gons), A358887, A006842, A006843, A005728, A358882.

Formula

a(n) = A358950(n) - A358948(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.

A359968 Number of vertices formed inside a right triangle by the straight line segments mutually connecting all vertices and points on the two shorter edges whose positions equal the Farey series of order n.

Original entry on oeis.org

3, 6, 37, 195, 1467, 3408, 17113, 40435, 109638, 191718, 572939, 842487, 2139708, 3231583, 5261013
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Jan 20 2023

Keywords

Comments

The number of vertices along the shorter edges is A005728(n). No formula for a(n) is known. The sequence is inspired by the Farey fan; see A360042.

Links

Crossrefs

Cf. A359969 (regions), A359970 (edges), A359971 (k-gons), A005728, A360042, A359974, A359690, A358949, A358887.

Formula

a(n) = A359970(n) - A359969(n) + 1 by Euler's formula.

A359974 Number of vertices formed inside a right triangle by the straight line segments mutually connecting all vertices and points on the two shorter edges whose positions on one edge equal the Farey series of order n while on the other they divide its length into n equal segments.

Original entry on oeis.org

3, 6, 26, 93, 424, 876, 2785, 5542, 11575, 18761, 40249, 57399, 109376, 155965, 227884, 322377, 532454, 676282, 1056010, 1334975, 1767798, 2240664, 3252047, 3882192, 5226897
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Jan 20 2023

Keywords

Comments

The number of vertices on the edge with point positions equaling the Farey series of order n is A005728(n). No formula for a(n) is known.
This graph is related to the 'Farey fan' given in the reference.

References

  • McIlroy, M. D. "A Note on Discrete Representation of Lines". AT&T Technical Journal, 64 (1985), 481-490.

Links

Crossrefs

Cf. A359975 (regions), A359976 (edges), A359977 (k-gons), A005728, A359968, A359690, A358949, A358887.

Formula

a(n) = A359976(n) - A359975(n) + 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 1-10 of 10 results.

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

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