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 1-5 of 5 results.

A359968 Number of vertices 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

3, 6, 37, 195, 1467, 3408, 17113, 40435, 109638, 191718, 572939, 842487, 2139708, 3231583, 5261013
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.

Links

Crossrefs

Cf. A359969 (regions), A359970 (edges), A359971 (k-gons), A005728, A360042, A359974, A359690, A358949, A358887.

Formula

a(n) = A359970(n) - A359969(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

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.

Links

Crossrefs

Cf. A359968 (vertices), A359970 (edges), A359971 (k-gons), A005728, A360042, A359975, A359690, A358948, A358886.

Formula

a(n) = A359970(n) - A359968(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.

A359976 Number of edges 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

3, 10, 55, 202, 902, 1868, 5886, 11676, 24322, 39440, 84155, 120151, 228121, 324856, 474396, 670552, 1104433, 1402237, 2185044, 2761367, 3654893, 4628608, 6706612, 8005739, 10770733
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.

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), A359977 (k-gons), A005728, A359970, A359693, A358950, A358888.

Formula

a(n) = A359974(n) + A359975(n) - 1 by Euler's formula.

A360043 Number of edges in a Farey fan of order n.

Original entry on oeis.org

4, 9, 18, 30, 52, 74, 112, 154, 210, 268, 352, 436, 552, 668, 802, 948, 1134, 1316, 1546, 1778, 2038, 2306, 2630, 2952, 3326, 3704, 4124, 4556, 5060, 5552, 6126, 6710, 7338, 7978, 8674, 9376, 10174, 10972, 11824, 12692, 13664, 14620, 15690, 16768, 17898, 19048, 20314, 21574, 22944, 24312
Offset: 1

Views

Author

Scott R. Shannon, N. J. A. Sloane and M. Douglas McIlroy, Jan 23 2023

Keywords

Comments

See the reference for the definition of a 'Farey fan'. See A360042 and A360044 for further details and images of the graph.

References

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

Crossrefs

Cf. A005598 (regions), A360042 (vertices), A360044 (k-gons), A005728, A174030, A359976, A359970, A359693.
Showing 1-5 of 5 results.

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

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