A358888 -id:A358888 - cp's OEIS frontend

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).

4, 56, 1040, 6064, 53104, 115496, 629920, 1457744, 3952264, 6835568

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).

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.

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.
Wikipedia, Farey sequence.

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.

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

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

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

			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;
.
.

Scott R. Shannon, Image for n = 5.
Wikipedia, Farey sequence.

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.

Sum of row n = A358886(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

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

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

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.
Wikipedia, Farey sequence.

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.

a(n) = A358888(n) - A358886(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.

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

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.
Wikipedia, Farey sequence.

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.

a(n) = A358882(n) + A358883(n) - 1 by Euler's formula.

A358950 Number of edges 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, 21, 375, 2574, 22083, 52791, 279750, 673050, 1851816, 3272058, 9865560, 14592537

Offset: 1

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

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

See A358948 and A358949 for images of the square.

Wikipedia, Farey sequence.

Cf. A358948 (regions), A358949 (vertices), A358951 (k-gons), A358888, A006842, A006843, A005728, A358882.

a(n) = A358948(n) + A358949(n) - 1 by Euler's formula.

A359970 Number of edges 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, 10, 84, 433, 3264, 7357, 37065, 86441, 232975, 405510, 1210898, 1773121, 4500787, 6774404, 10997356

Offset: 1

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

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.

See A359968 and A359969 for images of the triangle.

Cf. A359968 (vertices), A359969 (regions), A359971 (k-gons), A005728, A360042, A359976, A359693, A358950, A358888.

a(n) = A359968(n) + A359969(n) - 1 by Euler's formula.

A359976 Number of edges 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, 10, 55, 202, 902, 1868, 5886, 11676, 24322, 39440, 84155, 120151, 228121, 324856, 474396, 670552, 1104433, 1402237, 2185044, 2761367, 3654893, 4628608, 6706612, 8005739, 10770733

Offset: 1

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

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.
See A359974 and A359975 for images of the triangle.
This graph is related to the 'Farey fan' given in the reference.

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

Cf. A359974 (vertices), A359975 (regions), A359977 (k-gons), A005728, A359970, A359693, A358950, A358888.

a(n) = A359974(n) + A359975(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.

A359118 Number of edges in the planar Farey Ring graph FR(n) defined in A359116, including the regions between the convex hull and the bounding circle.

Original entry on oeis.org

1, 2, 12, 48, 457, 1027, 6190, 14652, 40852, 71601, 223637, 325661, 847984, 1269433, 2053303, 3157887, 6638971, 8490949, 16421392, 21323264, 30639928

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Dec 17 2022

See A359116 and A359117 for images of the figure.

Cf. A359116 (vertices), A359117 (regions), A359119 (k-gons), A358888, A006842, A006843, A005728, A135565.

a(n) = A359116 + A359117 - 1 by Euler's formula.

cp's OEIS Frontend

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).

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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).

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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.

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

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A358950 Number of edges 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).

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A359970 Number of edges 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.

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A359976 Number of edges 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.

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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.

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A359118 Number of edges in the planar Farey Ring graph FR(n) defined in A359116, including the regions between the convex hull and the bounding circle.

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