cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 17 results. Next

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

Views

Author

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

Keywords

Crossrefs

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

Views

Author

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

Keywords

Crossrefs

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.

Programs

  • Mathematica
    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 *)

A358882 The number of regions in a Farey diagram of order (n,n).

Original entry on oeis.org

4, 56, 504, 2024, 8064, 18200, 50736, 99248, 202688, 343256, 657904, 983008, 1708672, 2485968, 3755184, 5289944, 8069736, 10539792, 15387320, 19913840
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Dec 05 2022

Keywords

Comments

See A358298 and also 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.

Links

Crossrefs

Cf. A358883 (vertices), A358884 (edges), A358885 (k-gons), A006842, A006843, A005728, A358886.
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.

Formula

a(n) = A358884(n) - A358883(n) + 1 by Euler's formula.

A358886 Number of regions formed inside a square 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

4, 56, 1040, 6064, 53104, 115496, 629920, 1457744, 3952264, 6835568
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Dec 05 2022

Keywords

Comments

The number of points along each edge is given by A005728(n).
We call this graph the Completed Farey Diagram of order (n,k). The (ordinary) Farey diagram Farey(n,k) is a subgraph. In the latter graph, not all pairs of boundary points are joined by lines.

Links

Crossrefs

Cf. A358888 (edges), A358887 (vertices), A358889 (k-gons), A006842, A006843, A005728, A358882.
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.

Formula

a(n) = A358888(n) - A358887(n) + 1 by Euler's formula.

A358889 Table read by rows: T(n,k) = number of k-gons, k >= 3, formed inside a square 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

4, 48, 8, 712, 304, 24, 3368, 2400, 280, 16, 27424, 20360, 4784, 504, 32, 56000, 47088, 10912, 1400, 88, 8, 292424, 255608, 69368, 11504, 960, 56, 658800, 590208, 175856, 30160, 2496, 200, 24, 1748112, 1593912, 506496, 93584, 9616, 520, 24, 2981448, 2778456, 890368, 166912, 17192, 1144, 48
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Dec 05 2022

Keywords

Comments

The number of points along each edge is given by A005728(n).

Examples

			The table begins:
 4;
 48,      8;
 712,     304,     24;
 3368,    2400,    280,    16;
 27424,   20360,   4784,   504,    32;
 56000,   47088,   10912,  1400,   88,    8;
 292424,  255608,  69368,  11504,  960,   56;
 658800,  590208,  175856, 30160,  2496,  200,  24;
 1748112, 1593912, 506496, 93584,  9616,  520,  24;
 2981448, 2778456, 890368, 166912, 17192, 1144, 48;
.
.

Links

Crossrefs

Cf. A358886 (regions), A358887 (vertices), A358888 (edges), A006842, A006843, A005728, A358885.
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.

Formula

Sum of row n = A358886(n).

A358885 Table read by rows: T(n,k) = the number of regions with k sides, k >= 3, in a Farey diagram of order (n,n).

Original entry on oeis.org

4, 48, 8, 400, 104, 1568, 456, 6216, 1848, 13944, 4256, 38760, 11976, 75768, 23480, 154440, 48248, 261072, 82184, 500464, 157440, 747480, 235528, 1298584, 410088, 1890184, 595784, 2853416, 901768, 4015552, 1274392, 6127632, 1942104, 8002552, 2537240, 11683880, 3703440, 15123800, 4790040
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Dec 05 2022

Keywords

Comments

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.
It would be nice to have a proof (or disproof) that the number of sides is always 3 or 4.

Examples

			The table begins:
4;
48, 8;
400, 104;
1568, 456;
6216, 1848;
13944, 4256;
38760, 11976;
75768, 23480;
154440, 48248;
261072, 82184;
500464, 157440;
747480, 235528;
1298584, 410088;
1890184, 595784;
2853416, 901768;
4015552, 1274392;
6127632, 1942104;
8002552, 2537240;
11683880, 3703440;
15123800, 4790040;
.
.

Links

Crossrefs

Cf. A358882 (regions), A358883 (vertices), A358884 (edges), A006842, A006843, A005728, A358889.
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.

Formula

Sum of row n = A358882(n).

A358307 Main diagonal of array in A358304, divided by 2.

Original entry on oeis.org

0, 1, 5, 16, 33, 67, 102, 171, 241, 346, 452, 627, 769, 1015, 1228, 1512, 1796, 2220, 2541, 3072, 3500, 4070, 4605, 5386, 5958, 6848, 7598, 8561, 9419, 10665, 11525, 12950, 14094, 15524, 16812, 18528, 19818, 21852, 23463, 25467, 27187, 29687, 31409, 34160, 36310, 38890, 41255, 44544, 46840, 50347, 53037, 56477, 59489
Offset: 0

Views

Author

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

Keywords

Crossrefs

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.

A358887 Number of vertices formed inside a square 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

5, 37, 705, 4549, 42357, 94525, 531485, 1250681, 3440621, 5985201
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Dec 05 2022

Keywords

Comments

The number of points along each edge is given by A005728(n).

Links

Crossrefs

Cf. A358888 (edges), A358886 (regions), A358889 (k-gons), A006842, A006843, A005728, A358882, A358883.
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.

Formula

a(n) = A358888(n) - A358886(n) + 1 by Euler's formula.

A358888 Number of edges formed inside a square 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

8, 92, 1744, 10612, 95460, 210020, 1161404, 2708424, 7392884, 12820768
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Dec 05 2022

Keywords

Comments

The number of points along each edge is given by A005728(n).
See A358886 and A358887 for images of the square.

Links

Crossrefs

Cf. A358886 (regions), A358887 (vertices), A358889 (k-gons), A006842, A006843, A005728, A358882, A358884.
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.

Formula

a(n) = A358886(n) + A358887(n) - 1 by Euler's formula.

A358883 The number of vertices in a Farey diagram of order (n,n).

Original entry on oeis.org

5, 37, 313, 1253, 4977, 11253, 31393, 61409, 125525, 212785, 407757, 609361, 1059497, 1541005, 2328621, 3282329, 5006113, 6538721, 9545621, 12352197
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Dec 05 2022

Keywords

Comments

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.

Links

Crossrefs

Cf. A358882 (regions), A358884 (edges), A358885 (k-gons), A006842, A006843, A005728, A358887.
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.

Formula

a(n) = A358884(n) - A358882(n) + 1 by Euler's formula.
Showing 1-10 of 17 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.