A359968 -id:A359968 - 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.

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.

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.

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.

A359971 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 equal the Farey series of order n.

Original entry on oeis.org

1, 5, 33, 15, 108, 126, 5, 727, 1031, 38, 2, 1314, 2452, 167, 15, 2, 6811, 12102, 988, 52, 14904, 27626, 3255, 214, 4, 2, 2, 39172, 73289, 10062, 795, 19, 1, 65833, 127951, 18476, 1464, 64, 5, 201643, 370880, 59630, 5548, 250, 7, 2, 288196, 541258, 91037, 9692, 428, 20, 4, 741597, 1351301, 239180, 27510, 1434, 58

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.

			The table begins:
        1;
        5;
       33,      15;
      108,     126,      5;
      727,    1031,     38,     2;
     1314,    2452,    167,    15,    2;
     6811,   12102,    988,    52;
    14904,   27626,   3255,   214,    4,   2, 2;
    39172,   73289,  10062,   795,   19,   1;
    65833,  127951,  18476,  1464,   64,   5;
   201643,  370880,  59630,  5548,  250,   7, 2;
   288196,  541258,  91037,  9692,  428,  20, 4;
   741597, 1351301, 239180, 27510, 1434,  58;
  1095197, 2025237, 374907, 44880, 2491, 104, 4, 2;
  1747260, 3279178, 628335, 76787, 4600, 178, 6;
  ...

Cf. A359968 (vertices), A359969 (regions and row sums), A359970 (edges), A005728, A360042, A359977, A359694, A358951, A358889.

cp's OEIS Frontend

A360042 Number of vertices in a Farey fan 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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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.

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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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A359971 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 equal the Farey series of order n.

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