A358886 -id:A358886 - cp's OEIS frontend

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

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.

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.

A359117 Number of regions in the planar Farey Ring graph FR(n) defined in A359116, including the regions between the convex hull and the bounding circle.

Original entry on oeis.org

1, 2, 8, 30, 250, 548, 3180, 7468, 20684, 36164, 112406, 163600, 425144, 636208, 1028934, 1581766, 3323182, 4249976, 8216442, 10668790, 15329216

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Dec 17 2022

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.
Scott R. Shannon, Image for n = 8.
Scott R. Shannon, Image for n = 9.
Scott R. Shannon, Image for n = 10.

Cf. A359116 (vertices), A359118 (edges), A359119 (k-gons), A358886, A006842, A006843, A005728, A007678.

a(n) = A359118 - A359116 + 1 by Euler's formula.

A358304 Array read by antidiagonals: T(n,k) (n>=0, k>=0) = number of decreasing lines defining the Farey diagram Farey(n,k) of order (n,k).

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 0, 5, 5, 0, 0, 9, 10, 9, 0, 0, 14, 19, 19, 14, 0, 0, 20, 27, 32, 27, 20, 0, 0, 27, 40, 47, 47, 40, 27, 0, 0, 35, 51, 68, 66, 68, 51, 35, 0, 0, 44, 68, 85, 96, 96, 85, 68, 44, 0, 0, 54, 82, 112, 118, 134, 118, 112, 82, 54, 0, 0, 65, 103, 137, 156, 167, 167, 156, 137, 103, 65, 0, 0, 77, 120, 166, 187, 217, 204, 217, 187, 166, 120, 77, 0

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Dec 06 2022

			The full array T(n,k), n >= 0, k >= 0, begins:
  0,  0,  0,  0,   0,   0,   0,   0,   0,   0,   0,   0,   0, ..
  0,  2,  5,  9,  14,  20,  27,  35,  44,  54,  65,  77,  90, ..
  0,  5, 10, 19,  27,  40,  51,  68,  82, 103, 120, 145, 165, ..
  0,  9, 19, 32,  47,  68,  85, 112, 137, 166, 196, 235, 265, ..
  0, 14, 27, 47,  66,  96, 118, 156, 187, 229, 266, 320, 358, ..
  0, 20, 40, 68,  96, 134, 167, 217, 261, 317, 366, 436, 491, ..
  0, 27, 51, 85, 118, 167, 204, 267, 318, 384, 441, 528, 589, ..
  ...

Daniel Khoshnoudirad, Farey lines defining Farey diagrams and application to some discrete structures, Applicable Analysis and Discrete Mathematics, 9 (2015), 73-84; doi:10.2298/AADM150219008K. See Lemma 1, |DFD(m,n)|.

Cf. A358298.

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.

A005728 := proc(n) 1+add(numtheory[phi](i), i=1..n) ; end proc: # called F_n in the paper
Amn:=proc(m,n) local a,i,j;  # A331781 or equally A333295. Diagonal is A018805.
a:=0; for i from 1 to m do for j from 1 to n do
if igcd(i,j)=1 then a:=a+1; fi; od: od: a; end;
DFD:=proc(m,n) local d,t1,u,v; global A005728, Amn;
t1:=0; for u from 1 to m do for v from 1 to n do
d:=igcd(u,v); if d>=1 then t1:=t1 + (u+v)*numtheory[phi](d)/d; fi; od: od:
t1; end;
for m from 0 to 8 do lprint([seq(DFD(m,n),n=0..20)]); od:

T[n_, k_] := Sum[d = GCD[u, v]; If[d >= 1, (u+v)*EulerPhi[d]/d, 0], {u, 1, n}, {v, 1, k}];
Table[T[n-k, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 18 2023 *)

A358299 Triangle read by antidiagonals: T(n,k) (n>=0, 0 <= k <= n) = number of lines defining the Farey diagram of order (n,k).

Original entry on oeis.org

2, 3, 6, 4, 11, 20, 6, 19, 36, 60, 8, 29, 52, 88, 124, 12, 43, 78, 128, 180, 252, 14, 57, 100, 162, 224, 316, 388, 20, 77, 136, 216, 298, 412, 508, 652, 24, 97, 166, 266, 360, 498, 608, 780, 924, 30, 121, 210, 326, 444, 608, 738, 940, 1116, 1332, 34, 145, 246, 386, 518, 706, 852, 1086, 1280, 1532, 1748

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Dec 06 2022

			The full array T(n,k), n >= 0, k>= 0, begins:
2, 3, 4, 6, 8, 12, 14, 20, 24, 30, 34, 44, 48, 60, ...
3, 6, 11, 19, 29, 43, 57, 77, 97, 121, 145, 177, 205, ...
4, 11, 20, 36, 52, 78, 100, 136, 166, 210, 246, 302, ...
6, 19, 36, 60, 88, 128, 162, 216, 266, 326, 386, 468, ...
8, 29, 52, 88, 124, 180, 224, 298, 360, 444, 518, 628, ...
12, 43, 78, 128, 180, 252, 316, 412, 498, 608, 706, ...
14, 57, 100, 162, 224, 316, 388, 508, 608, 738, 852, ...
...

Daniel Khoshnoudirad, Farey lines defining Farey diagrams and application to some discrete structures, Applicable Analysis and Discrete Mathematics, 9 (2015), 73-84; doi:10.2298/AADM150219008K. See Theorem 1, |DF(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.

A005728 := proc(n) 1+add(numtheory[phi](i), i=1..n) ; end proc: # called F_n in the paper
Amn:=proc(m,n) local a,i,j; # A331781 or equally A333295. Diagonal is A018805.
a:=0; for i from 1 to m do for j from 1 to n do
if igcd(i,j)=1 then a:=a+1; fi; od: od: a; end;
# The present sequence is:
Dmn:=proc(m,n) local d,t1,u,v,a; global A005728, Amn;
a:=A005728(m)+A005728(n);
t1:=0; for u from 1 to m do for v from 1 to n do
d:=igcd(u,v); if d>=1 then t1:=t1 + (u+v)*numtheory[phi](d)/d; fi; od: od:
a+2*t1-2*Amn(m,n); end;
for m from 1 to 8 do lprint([seq(Dmn(m,n),n=1..20)]); od:

A358300 Row 1 of array in A358298.

Original entry on oeis.org

3, 6, 11, 19, 29, 43, 57, 77, 97, 121, 145, 177, 205, 243, 277, 315, 355, 405, 447, 503, 551, 605, 659, 727, 783, 853, 917, 989, 1057, 1143, 1211, 1303, 1383, 1469, 1553, 1647, 1731, 1841, 1935, 2037, 2133, 2255, 2351, 2479, 2587, 2701, 2815, 2955, 3067, 3207, 3327, 3461

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Dec 06 2022

nonn

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.

A358301 Main diagonal of array in A358298.

Original entry on oeis.org

2, 6, 20, 60, 124, 252, 388, 652, 924, 1332, 1748, 2428, 2988, 3948, 4788, 5908, 7028, 8692, 9964, 12052, 13748, 16004, 18124, 21204, 23476, 26996, 29972, 33788, 37196, 42124, 45548, 51188, 55732, 61412, 66532, 73348, 78484, 86548, 92956, 100924, 107772, 117692, 124556, 135476, 144036

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Dec 06 2022

nonn

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.

A005728[n_] := 1 + Sum[EulerPhi[i], {i, 1, n}];
Amn[m_, n_] := Sum[If[GCD[i, j] == 1, 1, 0], {i, 1, m}, {j, 1, n}];
Dmn[m_, n_] := A005728[m] + A005728[n] + 2 Sum[d = GCD[u, v]; If[d >= 1, (u+v)*EulerPhi[d]/d, 0], {u, 1, m}, {v, 1, n}] - 2*Amn[m, n];
Table[Dmn[n, n], {n, 0, 44}] (* Jean-François Alcover, Apr 18 2023, after Maple code in A358298 *)

cp's OEIS Frontend

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

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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.

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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.

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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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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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A359117 Number of regions in the planar Farey Ring graph FR(n) defined in A359116, including the regions between the convex hull and the bounding circle.

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A358304 Array read by antidiagonals: T(n,k) (n>=0, k>=0) = number of decreasing lines defining the Farey diagram Farey(n,k) of order (n,k).

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A358299 Triangle read by antidiagonals: T(n,k) (n>=0, 0 <= k <= n) = number of lines defining the Farey diagram of order (n,k).

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A358300 Row 1 of array in A358298.

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A358301 Main diagonal of array in A358298.

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