A359974 -id:A359974 - cp's OEIS frontend

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

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.

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.

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.

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.

A359977 Irregular table read by rows: T(n,k) is the number of k-gons, k>=3, 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, 20, 8, 2, 50, 57, 3, 169, 274, 31, 5, 303, 646, 41, 2, 1, 889, 2011, 179, 21, 2, 1685, 4025, 388, 33, 4, 3466, 8283, 925, 67, 7, 5624, 13442, 1498, 106, 9, 1, 11896, 27907, 3718, 354, 30, 2, 16976, 40100, 5182, 461, 33, 1, 32506, 73806, 11249, 1118, 61, 6, 46187, 104453, 16380, 1747, 123, 1, 1

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.

			The table begins:
1;
5;
20, 8, 2;
50, 57, 3;
169, 274, 31, 5;
303, 646, 41, 2, 1;
889, 2011, 179, 21, 2;
1685, 4025, 388, 33, 4;
3466, 8283, 925, 67, 7;
5624, 13442, 1498, 106, 9, 1;
11896, 27907, 3718, 354, 30, 2;
16976, 40100, 5182, 461, 33, 1;
32506, 73806, 11249, 1118, 61, 6;
46187, 104453, 16380, 1747, 123, 1, 1;
67117, 152534, 24159, 2511, 181, 10, 1;
95276, 213798, 34962, 3824, 295, 21;
.
.

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

Cf. A359974 (vertices), A359975 (regions), A359976 (edges), A005728, A359971, A359694, A358951, A358889.

Sum of row n = A359975(n).

cp's OEIS Frontend

A360042 Number of vertices in a Farey fan 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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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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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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