A006843 -id:A006843 - 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).

A177407 Form triangle of weighted Farey fractions; read denominators by rows.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 1, 5, 7, 3, 9, 9, 3, 7, 5, 1, 1, 7, 11, 5, 17, 19, 7, 17, 13, 3, 5, 7, 9, 27, 27, 9, 7, 5, 3, 13, 17, 7, 19, 17, 5, 11, 7, 1, 1, 9, 15, 7, 25, 29, 11, 27, 21, 5, 9, 13, 17, 53, 55, 19, 15, 11, 7, 31, 41, 17, 47, 43, 13, 29, 19, 3, 11, 13, 5, 17, 19, 7, 23, 25, 9

Offset: 0

N. J. A. Sloane, Dec 10 2010

Start with the list of fractions 0/1, 1/1 and repeatedly insert the weighted mediants (2a+c)/(2b+d) and (a+2c)/(b+2d) between every pair of adjacent elements a/b and c/d of the list. The fractions are to be reduced before the insertion step.

			Triangle begins:
  0 1
  - -
  1 1
.
  0 1 2 1
  - - - -
  1 3 3 1
.
  0 1 2 1 4 5 2 5 4 1
  - - - - - - - - - -
  1 5 7 3 9 9 3 7 5 1

James Propp, Posting to the Math Fun Mailing List, Dec 10 2010.

Nathaniel Johnston, Table of n, a(n) for n = 0..29533 (first 10 rows of triangle)
Dhroova Aiylam and Tanya Khovanova, Weighted Mediants and Fractals, arXiv:1711.01475 [math.NT], 2017.

Cf. A177405, A177903, A006842/A006843.

a(44)-a(80) and some corrected terms from Nathaniel Johnston, Apr 12 2011

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.

A177405 Form triangle of weighted Farey fractions; read numerators by rows.

Original entry on oeis.org

0, 1, 0, 1, 2, 1, 0, 1, 2, 1, 4, 5, 2, 5, 4, 1, 0, 1, 2, 1, 4, 5, 2, 5, 4, 1, 2, 3, 4, 13, 14, 5, 4, 3, 2, 9, 12, 5, 14, 13, 4, 9, 6, 1, 0, 1, 2, 1, 4, 5, 2, 5, 4, 1, 2, 3, 4, 13, 14, 5, 4, 3, 2, 9, 12, 5, 14, 13, 4, 9, 6, 1, 4, 5, 2, 7, 8, 3, 10, 11

Offset: 0

N. J. A. Sloane, Dec 10 2010

Start with the list of fractions 0/1, 1/1 and repeatedly insert the weighted mediants (2a+c)/(2b+d) and (a+2c)/(b+2d) between every pair of adjacent elements a/b and c/d of the list. The fractions are to be reduced before the insertion step.

James Propp asks: Does every fraction between 0 and 1 with odd denominator appear in the triangle?

			Triangle begins:
0 1
- -
1 1
0 1 2 1
- - - -
1 3 3 1
0 1 2 1 4 5 2 5 4 1
- - - - - - - - - -
1 5 7 3 9 9 3 7 5 1
0 1 .2 1 .4 .5 2 .5 .4 1 2 3 4 13 14 5 4 3 2 .9 12 5 14 13 4 .9 6 1
- - -- - -- -- - -- -- - - - - -- -- - - - - -- -- - -- -- - -- - -
1 7 11 5 17 19 7 17 13 3 5 7 9 27 27 9 7 5 3 13 17 7 19 17 5 11 7 1

James Propp, Posting to the Math Fun Mailing List, Dec 10 2010.

Nathaniel Johnston, Table of n, a(n) for n = 0..29533 (first 10 rows of triangle)
Dhroova Aiylam, Tanya Khovanova, Weighted Mediants and Fractals, arXiv:1711.01475 [math.NT], 2017.

Cf. A177407, A177903, A006842/A006843.

Mma code from James Propp:
        Lengthen[L_] :=
         Module[{i, M}, M = Table[0, {3 Length[L]}];
          M[[1]] = Numerator[L[[1]]]/(2 + Denominator[L[[1]]]);
          M[[2]] = 2*Numerator[L[[1]]]/(1 + 2 Denominator[L[[1]]]);
          For[i = 1, i < Length[L], i++, M[[3 i]] = L[[i]];
           M[[3 i + 1]] = (2 Numerator[L[[i]]] +
               Numerator[L[[i + 1]]])/(2 Denominator[L[[i]]] +
               Denominator[L[[i + 1]]]);
           M[[3 i + 2]] = (Numerator[L[[i]]] +
               2 Numerator[L[[i + 1]]])/(Denominator[L[[i]]] +
               2 Denominator[L[[i + 1]]])]; M[[3 Length[L]]] = L[[Length[L]]];
           Return[M]]
        WF[n_] := WF[n] = If[n == 0, {1}, Lengthen[WF[n - 1]]]

a(45)-a(80) and some corrected terms from Nathaniel Johnston, Apr 12 2011

A213544 Sum of numerators of Farey Sequence of order n.

Original entry on oeis.org

1, 2, 5, 9, 19, 25, 46, 62, 89, 109, 164, 188, 266, 308, 368, 432, 568, 622, 793, 873, 999, 1109, 1362, 1458, 1708, 1864, 2107, 2275, 2681, 2801, 3266, 3522, 3852, 4124, 4544, 4760, 5426, 5768, 6236, 6556, 7376, 7628, 8531, 8971, 9511, 10017, 11098, 11482

Offset: 1

Anunay Kulshrestha, Jun 14 2012

			For n = 3, the Farey Sequence is 0/1, 1/3, 1/2, 2/3, 1/1. Thus a(3) = 0 + 1 + 1 + 2 + 1 = 5.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Wikipedia, Farey Sequence

Similar to A133404 and A191607.
Partial sums of A023896.
Cf. A006842, A006843, A011755, A240877.

with(numtheory):
b:= n-> `if`(n=1, 1, n*phi(n)/2):
a:= proc(n) option remember; b(n) +`if`(n>1, a(n-1), 0) end:
seq(a(n), n=1..60);  # Alois P. Heinz, Jun 14 2012

Farey[n_] := Union[ Flatten[ Join[{0}, Table[a/b, {b, n}, {a, b}]]]]; Table[ Total[ Numerator[ Farey[ n]]], {n, 0, 53}] (* Robert G. Wilson v, Apr 15 2014 *)
a[n_] := Sum[If[CoprimeQ[j, k], j, 0], {k, 1, n}, {j, 1, k}]; Table[a[n], {n, 1, 48}] (* Jean-François Alcover, Dec 29 2014 *)
Table[Total[Numerator[FareySequence[n]]],{n,50}] (* Harvey P. Dale, Apr 21 2025 *)

a(n) = Sum_{k=1..n} A023896(k).
a(n) = A240877(n)/2. - Robert G. Wilson v, Apr 15 2014
a(n) ~ n^3/Pi^2 - Jean-François Alcover, Dec 29 2014
a(n) = (A011755(n)+1)/2. - Chai Wah Wu, Apr 04 2022

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.

A359653 Number of regions 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

1, 4, 96, 728, 7840, 17744, 104136, 246108, 681704, 1187200, 3719496, 5396692, 14149896

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Jan 09 2023

The number of points internal to each edge is given by A005728(n) - 2.

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. A359654 (vertices), A359655 (edges), A359656 (k-gons), A005728, A358886, A358882, A355798, A358948, A006842, A006843.

a(n) = A359655(n) - A359654(n) + 1 by Euler's formula.

A359654 Number of vertices 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, 9, 77, 593, 6749, 15569, 93281, 222933, 623409, 1087393, 3453289, 5011009, 13271517

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.

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. A359653 (regions), A359655 (edges), A359656 (k-gons), A005728, A358887, A358883, A355799, A358949, A006842, A006843.

a(n) = A359655(n) - A359653(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.

A178031 Consider the Farey tree A049455/A049456; a(n) = row at which the denominator n first appears (assumes first row is labeled row 1).

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 7, 8, 8, 8, 8, 8, 8, 9, 8, 8, 8, 9, 9, 8, 9, 9, 9, 10, 9, 9, 9, 10, 9, 9, 9, 9, 9, 10, 9, 9, 10, 10, 10, 11, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 1

N. J. A. Sloane, Dec 16 2010

nonn

Computed by Alan Wechsler, Dec 16 2010.
Richard C. Schroeppel also asked about the analogous sequence giving the last occurrence of denominator n.
The first occurrence of k in this sequence is apparently at n = A135510(k-1), except for k=5. The last occurrence of k is at n = Fibonacci(k). - Andrey Zabolotskiy, Dec 01 2024

			Start with a pair of fractions 0/1, 1/1 and repeatedly insert the "Farey sum" (p+r)/(q+s) in between every pair of adjacent fractions p/q, r/s. The first few iterations are:
1:   0/1                                     1/1
2:   0/1                 1/2                 1/1
3:   0/1       1/3       1/2       2/3       1/1
4:   0/1  1/4  1/3  2/5  1/2  3/5  2/3  3/4  1/1
We only look at the denominators in this table (which form the sequence A049456, or A002487 if the rightmost column is removed).
1 first appears in row 1, so a(1) = 1.
2 first appears in row 2, so a(2) = 2.
3 first appears in row 3, so a(3) = 3.
4 and 5 first appear in row 4, so a(4) = a(5) = 4.

Based on a posting by Richard C. Schroeppel to the Math Fun Mailing List, Dec 15 2010.

Bo Gyu Jeong, Table of n, a(n) for n = 1..10000
Richard J. Mathar, The Kepler binary tree of reduced fractions, 2017.

See A178047 for another version. Cf. A002487, A006842, A006843, A177903, A178042, A135510.

More terms from Bo Gyu Jeong, Oct 20 2012

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A177407 Form triangle of weighted Farey fractions; read denominators by rows.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Extensions

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A177405 Form triangle of weighted Farey fractions; read numerators by rows.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Extensions

A213544 Sum of numerators of Farey Sequence of order n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A359653 Number of regions 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.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A359654 Number of vertices 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.

Views

Author

Keywords

Comments

Links

Crossrefs

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.

Views

Author

Keywords

Comments