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 1-6 of 6 results.

A358948 Number of regions 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

1, 12, 228, 1464, 12516, 29022, 153564, 364650, 996672, 1750326, 5274156, 7761498
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Dec 07 2022

Keywords

Comments

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

Links

Crossrefs

Cf. A358949 (vertices), A358950 (edges), A358951 (k-gons), A358886, A006842, A006843, A005728, A358882.

Formula

a(n) = A358950(n) - A358949(n) + 1 by Euler's formula.

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

Views

Author

Scott R. Shannon and N. J. A. Sloane, Dec 07 2022

Keywords

Comments

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

Links

Crossrefs

Cf. A358948 (regions), A358950 (edges), A358951 (k-gons), A358887, A006842, A006843, A005728, A358882.

Formula

a(n) = A358950(n) - A358948(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

Views

Author

Scott R. Shannon and N. J. A. Sloane, Dec 07 2022

Keywords

Comments

The number of points along each edge is given by A005728(n).
See A358948 and A358949 for images of the square.

Links

Crossrefs

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

Formula

a(n) = A358948(n) + A358949(n) - 1 by Euler's formula.

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.

Original entry on oeis.org

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

Views

Author

Scott R. Shannon and N. J. A. Sloane, Jan 20 2023

Keywords

Comments

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.

Examples

			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;
  ...

Crossrefs

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

Views

Author

Scott R. Shannon and N. J. A. Sloane, Jan 20 2023

Keywords

Comments

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.

Examples

			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;
.
.

References

  • McIlroy, M. D. "A Note on Discrete Representation of Lines". AT&T Technical Journal, 64 (1985), 481-490.

Crossrefs

Cf. A359974 (vertices), A359975 (regions), A359976 (edges), A005728, A359971, A359694, A358951, A358889.

Formula

Sum of row n = A359975(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

Views

Author

Scott R. Shannon and N. J. A. Sloane, Jan 10 2023

Keywords

Comments

The number of points internal to each edge is given by A005728(n) - 2.
See A359653 and A359654 for images of the square.

Examples

			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;
.
.

Crossrefs

Cf. A359653 (regions), A359654 (vertices), A359655 (edges), A005728, A358889, A358885, A355801, A358951, A006842, A006843.
Showing 1-6 of 6 results.

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

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