A358889 -id:A358889 - cp's OEIS frontend

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.

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.

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).

A359119 Irregular table read by rows: T(n,k) is the number of k-gons, k>=2, in the Farey Ring graph FR(n) defined in A359116.

Original entry on oeis.org

2, 4, 4, 6, 18, 6, 10, 124, 76, 32, 8, 12, 244, 196, 78, 14, 4, 18, 1184, 1296, 534, 118, 28, 2, 22, 2632, 3180, 1244, 330, 58, 2, 28, 7244, 8628, 3594, 962, 190, 38, 32, 12626, 14922, 6378, 1836, 330, 36, 4, 42, 39060, 45656, 20152, 6082, 1252, 132, 28, 2, 46, 56980, 66088, 29454, 8916, 1840, 244, 26, 6

Offset: 2

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

See A359116 and A359117 for further images of the graphs.

			The table begins:
2;
4, 4;
6, 18, 6;
10, 124, 76, 32, 8;
12, 244, 196, 78, 14, 4;
18, 1184, 1296, 534, 118, 28, 2;
22, 2632, 3180, 1244, 330, 58, 2;
28, 7244, 8628, 3594, 962, 190, 38;
32, 12626, 14922, 6378, 1836, 330, 36, 4;
42, 39060, 45656, 20152, 6082, 1252, 132, 28, 2;
46, 56980, 66088, 29454, 8916, 1840, 244, 26, 6;
58, 148058, 170352, 76834, 23936, 4990, 766, 136, 12, 0, 2;
64, 221744, 253808, 115806, 35922, 7428, 1286, 136, 14;
72, 359676, 408252, 188438, 57828, 12432, 1972, 246, 16, 2;
80, 553598, 624588, 291158, 89762, 19066, 3104, 374, 30, 6;
96, 1164192, 1305260, 615048, 189910, 41094, 6654, 844, 72, 12;
102, 1491314, 1664362, 788138, 243924, 52438, 8502, 1080, 112, 2, 2;
120, 2887184, 3203244, 1529870, 474822, 102482, 16490, 2002, 206, 22;
128, 3752194, 4153544, 1987610, 617634, 133288, 21374, 2698, 278, 42;
140, 5393824, 5962776, 2855524, 889822, 191612, 31128, 3926, 438, 26;
.
.

Scott R. Shannon, Image for n = 11.

Cf. A359116 (vertices), A359117 (regions), A359118 (edges), A358889, A006842, A006843, A005728, A331451.

Sum of row n = A359117(n).

A359656 Irregular table read by rows: T(n,k) is the number of k-gons, k>=3, 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

0, 1, 0, 4, 56, 40, 368, 300, 48, 12, 3376, 3408, 960, 96, 7536, 7524, 2240, 436, 8, 42048, 45112, 13912, 2868, 168, 28, 97720, 105980, 34496, 7020, 832, 52, 8, 267240, 290456, 100560, 20576, 2688, 160, 24, 461800, 509824, 174400, 36228, 4608, 324, 16, 1411272, 1594296, 569152, 126408, 16856, 1408, 104

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.

See A359653 and A359654 for images of the square.

			The table begins:
0, 1;
0, 4;
56, 40;
368, 300, 48, 12;
3376, 3408, 960, 96;
7536, 7524, 2240, 436, 8;
42048, 45112, 13912, 2868, 168, 28;
97720, 105980, 34496, 7020, 832, 52, 8;
267240, 290456, 100560, 20576, 2688, 160, 24;
461800, 509824, 174400, 36228, 4608, 324, 16;
1411272, 1594296, 569152, 126408, 16856, 1408, 104;
2054616, 2300184, 830280, 184664, 24480, 2332, 128, 8;
5296752, 6001228, 2253456, 517564, 72888, 7532, 472, 4;
.
.

Cf. A359653 (regions), A359654 (vertices), A359655 (edges), A005728, A358889, A358885, A355801, A358951, A006842, A006843.

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 *)

A358302 Number of triangular regions in the Farey Diagram Farey(n,n), divided by 4.

Original entry on oeis.org

1, 12, 100, 392, 1554, 3486, 9690, 18942, 38610, 65268, 125116, 186870, 324646, 472546, 713354, 1003888, 1531908, 2000638, 2920970, 3780950

Offset: 1

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

This is the leading column in A358885, divided by 4.

It would be nice to have a formula.

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.

A358303 Number of 4-sided regions in the Farey Diagram Farey(n,n), divided by 8.

Original entry on oeis.org

1, 13, 57, 231, 532, 1497, 2935, 6031, 10273, 19680, 29441, 51261, 74473, 112721, 159299, 242763, 317155, 462930, 598755

Offset: 1

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

This is the second column in A358885, divided by 8.

It would be nice to have a formula.

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.

cp's OEIS Frontend

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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A359119 Irregular table read by rows: T(n,k) is the number of k-gons, k>=2, in the Farey Ring graph FR(n) defined in A359116.

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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

Mathematica

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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Maple

A358300 Row 1 of array in A358298.

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A358301 Main diagonal of array in A358298.

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Mathematica

A358302 Number of triangular regions in the Farey Diagram Farey(n,n), divided by 4.

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A358303 Number of 4-sided regions in the Farey Diagram Farey(n,n), divided by 8.

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