A358882 -id:A358882 - cp's OEIS frontend

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).

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.

A358950 Number of edges 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, 21, 375, 2574, 22083, 52791, 279750, 673050, 1851816, 3272058, 9865560, 14592537

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).

See A358948 and A358949 for images of the square.

Wikipedia, Farey sequence.

Cf. A358948 (regions), A358949 (vertices), A358951 (k-gons), A358888, A006842, A006843, A005728, A358882.

a(n) = A358948(n) + A358949(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.

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

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

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A358950 Number of edges 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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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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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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Maple

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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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