A359690 -id:A359690 - cp's OEIS frontend

A359694 Irregular table read by rows: T(n,k) is the number of k-gons, k>=3, in a regular drawing of a complete bipartite graph where the vertex positions on each part equal the Farey series of order n.

2, 10, 2, 70, 24, 218, 160, 4, 1254, 1068, 148, 16, 2254, 2414, 252, 26, 10082, 11760, 1980, 266, 12, 21410, 25958, 5096, 648, 36, 4, 53422, 68208, 14360, 1980, 168, 20, 86986, 118922, 24028, 3056, 248, 12, 0, 2, 255678, 346676, 84344, 12774, 1132, 110, 4, 2, 365674, 493530, 119820, 18600, 1624, 112, 4

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 is known.

See A359692 for other images of the graph.

			The table begins:
2;
10, 2;
70, 24;
218, 160, 4;
1254, 1068, 148, 16;
2254, 2414, 252, 26;
10082, 11760, 1980, 266, 12;
21410, 25958, 5096, 648, 36, 4;
53422, 68208, 14360, 1980, 168, 20;
86986, 118922, 24028, 3056, 248, 12, 0, 2;
255678, 346676, 84344, 12774, 1132, 110, 4, 2;
365674, 493530, 119820, 18600, 1624, 112, 4;
917478, 1244492, 334096, 57080, 5700, 478, 16, 4;
1335398, 1862666, 495536, 82642, 8096, 676, 24, 6;
2107042, 2989864, 788340, 128378, 12536, 932, 52, 4;
3195474, 4557430, 1230300, 205352, 20516, 1664, 80, 4;
.
.

Scott R. Shannon, Image for n = 7.
Eric Weisstein's World of Mathematics, Complete Bipartite Graph.
Wikipedia, Farey sequence.

Cf. A359690 (vertices), A359691 (crossings), A359692 (regions), A359693 (edges), A005728, A290131, A359653, A358886, A358882, A006842, A006843.

Sum of row n = A359692(n).

A360042 Number of vertices in a Farey fan of order n.

Original entry on oeis.org

4, 6, 11, 17, 29, 39, 59, 79, 107, 133, 175, 213, 271, 323, 385, 451, 541, 621, 731, 835, 955, 1073, 1225, 1367, 1541, 1707, 1897, 2087, 2321, 2535, 2801, 3061, 3345, 3625, 3937, 4243, 4609, 4957, 5335, 5713, 6155, 6569, 7055, 7529, 8031, 8531, 9101, 9649, 10265, 10859

Offset: 1

Scott R. Shannon, N. J. A. Sloane and M. Douglas McIlroy Jan 23 2023

nonn

See the reference for the definition of a 'Farey fan'.
The number of vertices along each edge is A005728(n), while the number of regions is conjectured to equal A005598(n) = 1 + Sum_{i=1..n} (n-i+1)*phi(i). The regions count the number of distinct approximate representations of straight lines y = mx + b that can be drawn on an x-y integer raster, where x, y, and b are restricted to [0,n) and 0 <= m <=1.
It is also worth noting that for 3 <= n <= 10 this sequence equals 2*A005728(n) + A174030(n-2), where A174030(n) = Sum_{i=1..n} (i where phi(i)|i). That is, the number of internal vertices of the Farey fan equals A174030(n) in this range. This may suggest a possible attack on finding a formula for the present sequence.

M. Douglas McIlroy, A Note on Discrete Representation of Lines, AT&T Technical Journal, 64 (1985), 481-490.
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 = 10.

Cf. A005598 (regions), A360043 (edges), A360044 (k-gons), A005728, A174030, A359974, A359968, A359690.

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.

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

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.

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.

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

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

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.

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.

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.

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

a(n) = A359976(n) - A359975(n) + 1 by Euler's formula.

A359692 Number of regions 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

2, 12, 94, 382, 2486, 4946, 24100, 53152, 138158, 233254, 700720, 999364, 2559344, 3785044, 6027148, 9210820

Offset: 1

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

nonn

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

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

Cf. A359690 (vertices), A359691 (crossings), A359693 (edges), A359694 (k-gons), A005728, A290131, A359653, A358886, A358882, A006842, A006843.

a(n) = A359693(n) - A359690(n) + 1 by Euler's formula.

A359969 Number of regions 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

1, 5, 48, 239, 1798, 3950, 19953, 46007, 123338, 213793, 637960, 930635, 2361080, 3542822, 5736344

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.

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.

Cf. A359968 (vertices), A359970 (edges), A359971 (k-gons), A005728, A360042, A359975, A359690, A358948, A358886.

a(n) = A359970(n) - A359968(n) + 1 by Euler's formula.

A359975 Number of regions 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

1, 5, 30, 110, 479, 993, 3102, 6135, 12748, 20680, 43907, 62753, 118746, 168892, 246513, 348176, 571980, 725956, 1129035, 1426393, 1887096, 2387945, 3454566, 4123548, 5543837

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.

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.

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.

Cf. A359974 (vertices), A359976 (edges), A359977 (k-gons), A005728, A359969, A359690, A358948, A358886.

a(n) = A359976(n) - A359974(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

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 for images of the graph.

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

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

a(n) = A359690(n) - 2*A005728(n).

cp's OEIS Frontend

A359694 Irregular table read by rows: T(n,k) is the number of k-gons, k>=3, 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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A360042 Number of vertices in a Farey fan of order 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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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.

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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.

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A359692 Number of regions 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

A359969 Number of regions 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.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A359975 Number of regions 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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Author

Keywords

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References

Links

Crossrefs

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula