A358950 -id:A358950 - cp's OEIS frontend

A358948 Number of regions 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

1, 12, 228, 1464, 12516, 29022, 153564, 364650, 996672, 1750326, 5274156, 7761498

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).

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.
N. J. A. Sloane, New Gilbreath Conjectures, Sum and Erase, Dissecting Polygons, and Other New Sequences, Doron Zeilberger's Exper. Math. Seminar, Rutgers, Sep 14 2023: Video, Slides, Updates. (Mentions this sequence.)
Wikipedia, Farey sequence.

Cf. A358949 (vertices), A358950 (edges), A358951 (k-gons), A358886, A006842, A006843, A005728, A358882.

a(n) = A358950(n) - A358949(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

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

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

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.
N. J. A. Sloane, New Gilbreath Conjectures, Sum and Erase, Dissecting Polygons, and Other New Sequences, Doron Zeilberger's Exper. Math. Seminar, Rutgers, Sep 14 2023: Video, Slides, Updates. (Mentions this sequence.)
Wikipedia, Farey sequence.

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

a(n) = A358950(n) - A358948(n) + 1 by Euler's formula.

A358951 Irregular table read by rows: T(n,k) = number of k-gons, k >= 3, 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,m)/A006843(n,m), m = 1..A005728(n).

Original entry on oeis.org

1, 12, 180, 42, 6, 810, 576, 72, 6, 6786, 4932, 744, 48, 6, 13662, 12522, 2568, 258, 12, 72582, 64932, 14376, 1632, 36, 6, 164484, 155088, 38688, 5958, 414, 18, 439524, 422370, 114804, 18462, 1392, 120, 750108, 749928, 211518, 35226, 3336, 204, 6, 2265462, 2240994, 647184, 109602, 10230, 666, 18

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).

			The table begins:
1;
12;
180, 42, 6;
810, 576, 72, 6;
6786, 4932, 744, 48, 6;
13662, 12522, 2568, 258, 12;
72582, 64932, 14376, 1632, 36, 6;
164484, 155088, 38688, 5958, 414, 18;
439524, 422370, 114804, 18462, 1392, 120;
750108, 749928, 211518, 35226, 3336, 204, 6;
2265462, 2240994, 647184, 109602, 10230, 666, 18;
3263436, 3312270, 990072, 176172, 18294, 1188, 66;
.
.

Scott R. Shannon, Image for n = 7.
Wikipedia, Farey sequence.

Cf. A358948 (regions), A358949 (vertices), A358950 (edges), A358889, A006842, A006843, A005728, A358882.

Sum of row n = A358948(n).

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.

cp's OEIS Frontend

A358948 Number of regions 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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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).

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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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