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.

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A104431 Number of ways to split 1, 2, 3, ..., 5n into n arithmetic progressions each with 5 terms.

Original entry on oeis.org

1, 1, 2, 4, 10, 21, 59, 125, 349, 848, 2224, 5210, 15720, 37096, 98241, 245251, 684475, 1703174, 4915084, 12024901, 33594399
Offset: 0

Views

Author

Jonas Wallgren, Mar 17 2005

Keywords

Crossrefs

Cf. A104429-A104430, A104432-A104443, A349430.

Extensions

a(11)-a(18) from Alois P. Heinz, Dec 28 2011
a(19)-a(20) from Alois P. Heinz, Nov 18 2021

A104442 Number of ways to split 1, 2, 3, ..., tn into n arithmetic progressions each with t terms, t>n.

Original entry on oeis.org

1, 1, 2, 4, 10, 20, 56, 116, 320, 736, 1872, 4176, 12712, 28368, 73592, 177272, 487304, 1130712, 3209792, 7494720, 20419744, 49706280, 129803592, 311179624
Offset: 0

Views

Author

Jonas Wallgren, Mar 17 2005

Keywords

Crossrefs

Cf. A104429-A104441, A104443.

Extensions

a(0), a(13)-a(23) from Alois P. Heinz, Nov 18 2020

A279198 Number of pairs of conjugate inseparable solutions of X + Y = 2Z (integer, disjoint triples from {1,2,3,...,3n}).

Original entry on oeis.org

0, 0, 0, 2, 7, 52, 297, 1994, 14594, 113794, 991741, 9199390, 94105010, 1015012796, 11914379971, 146974330141, 1954701366709
Offset: 1

Views

Author

N. J. A. Sloane, Dec 15 2016

Keywords

Examples

			Richard Guy gives examples in his letter.

References

  • R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, Univ. Calgary, Dept. Mathematics, Research Paper No. 129, 1971.
  • R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, in Proc. Conf. Number Theory. Pullman, WA, 1971, pp. 221-223.
  • R. K. Guy, Packing [1,n] with solutions of ax + by = cz; the unity of combinatorics, in Colloq. Internaz. Teorie Combinatorie. Rome, 1973, Atti Conv. Lincei. Vol. 17, Part II, pp. 173-179, 1976.
  • Nowakowski, Richard Joseph, Generalization of the Langford-Skolem problem, MS Thesis, University of Calgary, 1975.

Links

  • R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [Annotated scanned copy, with permission] See sequence "J".
  • R. J. Nowakowski, Generalizations of the Langford-Skolem problem, M.S. Thesis, Dept. Math., Univ. Calgary, May 1975. [Scanned copy, with permission.]

Crossrefs

All of A279197, A279198, A202705, A279199, A104429, A282615 are concerned with counting solutions to X+Y=2Z in various ways.
See also A002848, A002849.

Formula

A279197(n) + 2*A279198(n) = A202705(n).

Extensions

a(7)-a(16) from Fausto A. C. Cariboni, Feb 27 2017
a(17) from Fausto A. C. Cariboni, Mar 22 2017

A282616 Number of self-conjugate solutions of X + Y = 2Z (integer, disjoint triples from {1,2,3,...,3n}).

Original entry on oeis.org

1, 2, 3, 5, 15, 20, 75, 93, 588, 602, 4954, 4854, 51068, 48779, 597554, 567644, 8039742, 7634924, 120721322, 114398957, 2017517155, 1889828995, 36749338386, 34451341024, 726198499999, 679116640274, 15459385244039, 14509756794668, 356501015466981, 332645434167718, 8701627694048482
Offset: 1

Views

Author

Peter Kagey, Feb 19 2017

Keywords

Comments

A self-conjugate solution is one in which for every triple (a, b, c) in the partition there exists a "conjugate" triple (m-a, m-b, m-c) or (m-b, m-a, m-c) where m = 3n+1.
| separable | inseparable | either |
-------------------+-----------+-------------+---------+
self-conjugate | A282615 | A279197 | A282616 |
non-self-conjugate | A282618 | A282617 | A282619 |
either | A279199 | A202705 | A104429 |

Examples

			For n = 3 the a(3) = 3 solutions are:
  (7,9,8),(4,6,5),(1,3,2),
  (3,9,6),(2,8,5),(1,7,4), and
  (6,8,7),(2,4,3),(1,9,5).

Crossrefs

Cf. A104429, A202705, A279197, A279199, A282615, A282617, A282618, A282619.

Formula

a(n) = A282615(n) + A279197(n).
a(n) = A104429(n) - A282619(n).

Extensions

a(11)-a(16) from Fausto A. C. Cariboni, Feb 27 2017
a(17) from Fausto A. C. Cariboni, Mar 22 2017
a(18)-a(24) from Bert Dobbelaere, May 29 2025
a(25)-a(31) from Martin Fuller, Jul 15 2025

A261516 Number of perfect rhythmic tilings of [0,3n-1] with triples.

Original entry on oeis.org

1, 1, 0, 0, 0, 2, 0, 18, 66, 382, 1104, 4138, 15324, 61644, 325456, 2320948, 17660110, 148271962, 1171109228, 9257051746
Offset: 0

Views

Author

Michel Marcus, Aug 23 2015

Keywords

Comments

A perfect tiling of the line with triples consists of groups of three evenly spaced points, each group having a different common interval such that all points of the line are covered.

Examples

			For n=1, there is 1 such tiling: (0,1,2).
For n=5, there are 2 such tilings: (2,3,4), (8,10,12), (5,9,13), (1,6,11), (0,7,14) and its mirror, that have these distinct common differences: 1,2,4,5,7.

References

  • J. P. Delahaye, La musique mathématique de Tom Johnson, in Mathématiques pour le plaisir, Belin-Pour la Science, Paris, 2010.

Links

Crossrefs

Cf. A060963, A104429, A261517, A285527.

Extensions

a(16)-a(17) from Alois P. Heinz, Sep 16 2015
a(18)-a(19) from Fausto A. C. Cariboni, Mar 27 2017
a(0)=1 prepended by Seiichi Manyama, Feb 21 2020

A282617 Number of non-self-conjugate inseparable solutions of X + Y = 2Z (integer, disjoint triples from {1,2,3,...,3n}).

Original entry on oeis.org

0, 0, 0, 4, 14, 104, 594, 3988, 29188, 227588, 1983482, 18398780, 188210020, 2030025592, 23828759942, 293948660282, 3909402733418, 54360500959634, 806312590045382
Offset: 1

Views

Author

Peter Kagey, Feb 19 2017

Keywords

Comments

An inseparable solution is one in which "there is no j such that the first j of the triples are a partition of 1, ..., 3j" (see A202705).
A self-conjugate solution is one in which for every triple (a, b, c) in the partition there exists a "conjugate" triple (m-a, m-b, m-c) or (m-b, m-a, m-c) where m = 3n+1.
| separable | inseparable | either |
-------------------+-----------+-------------+---------+
self-conjugate | A282615 | A279197 | A282616 |
non-self-conjugate | A282618 | A282617 | A282619 |
either | A279199 | A202705 | A104429 |

Examples

			For n = 4 the a(4) = 4 solutions are:
(7,11,9),(4,12,8),(2,10,6),(1,5,3),
(9,11,10),(4,8,6),(2,12,7),(1,5,3),
(8,12,10),(3,11,7),(2,6,4),(1,9,5), and
(8,12,10),(5,9,7),(2,4,3),(1,11,6).

Crossrefs

Cf. A104429, A202705, A279197, A279199, A282615, A282616, A282618, A282619.

Formula

a(n) = A282619(n) - A282618(n).
a(n) = A202705(n) - A279197(n).

Extensions

a(10)-a(16) from Fausto A. C. Cariboni, Feb 27 2017
a(17) from Fausto A. C. Cariboni, Mar 22 2017
a(18)-a(19) from Martin Fuller, Jul 15 2025

A282618 Number of non-self-conjugate separable solutions of X + Y = 2Z (integer, disjoint triples from {1,2,3,...,3n}).

Original entry on oeis.org

0, 0, 2, 6, 26, 108, 492, 2562, 14790, 98874, 720614, 5908394, 52572682, 516141316, 5422012074, 61889630476, 749456000504, 9767057565198, 134007980469502, 1958535740848524
Offset: 1

Views

Author

Peter Kagey, Feb 19 2017

Keywords

Comments

An inseparable solution is one in which "there is no j such that the first j of the triples are a partition of 1, ..., 3j" (see A202705).
A self-conjugate solution is one in which for every triple (a, b, c) in the partition there exists a "conjugate" triple (m-a, m-b, m-c) or (m-b, m-a, m-c) where m = 3n+1.
| separable | inseparable | either |
-------------------+-----------+-------------+---------+
self-conjugate | A282615 | A279197 | A282616 |
non-self-conjugate | A282618 | A282617 | A282619 |
either | A279199 | A202705 | A104429 |

Examples

			For n = 3 the a(3) = 2 solutions are:
(5,9,7),(4,8,6),(1,3,2), and
(7,9,8),(2,6,4),(1,5,3).

Crossrefs

Cf. A104429, A202705, A279197, A279199, A282615, A282616, A282617, A282619.

Formula

a(n) = A282619(n) - A282617(n).
a(n) = A279199(n) - A282615(n).

Extensions

a(10)-a(16) from Fausto A. C. Cariboni, Feb 27 2017
a(17) from Fausto A. C. Cariboni, Mar 22 2017
a(18)-a(20) from Martin Fuller, Jul 15 2025

A282619 Number of non-self-conjugate solutions of X + Y = 2Z (integer, disjoint triples from {1,2,3,...,3n}).

Original entry on oeis.org

0, 0, 2, 10, 40, 212, 1086, 6550, 43978, 326462, 2704096, 24307174, 240782702, 2546166908, 29250772016, 355838290758, 4658858733922, 64127558524832, 940320570514884
Offset: 1

Views

Author

Peter Kagey, Feb 19 2017

Keywords

Comments

A self-conjugate solution is one in which for every triple (a, b, c) in the partition there exists a "conjugate" triple (m-a, m-b, m-c) or (m-b, m-a, m-c) where m = 3n+1.
| separable | inseparable | either |
-------------------+-----------+-------------+---------+
self-conjugate | A282615 | A279197 | A282616 |
non-self-conjugate | A282618 | A282617 | A282619 |
either | A279199 | A202705 | A104429 |

Examples

			For n = 3 the a(3) = 3 solutions are
(5,9,7),(4,8,6),(1,3,2),
(7,9,8),(2,6,4),(1,5,3).

Crossrefs

Cf. A104429, A202705, A279197, A279199, A282615, A282616, A282617, A282618.

Formula

a(n) = A282617(n) + A282618(n).
a(n) = A104429(n) - A282616(n).

Extensions

a(11)-a(16) from Fausto A. C. Cariboni, Feb 27 2017
a(17) from Fausto A. C. Cariboni, Mar 22 2017
a(18)-a(19) from Martin Fuller, Jul 15 2025

A332748 The number of permutations of {1,1,1,2,2,2,...,n,n,n} such that each triple of k's (k=1..n) is equally spaced with b(k) other elements in between and b(1) >= b(2) >= ... >= b(n).

Original entry on oeis.org

1, 1, 4, 18, 124, 738, 7464, 55890, 668778, 7030210, 90713844, 1054221258, 18597735744, 242795838520
Offset: 0

Views

Author

Seiichi Manyama, Feb 21 2020

Keywords

Examples

			n = 1 case:
     |           | b(1)
-----+-----------+------
   1 | [1, 1, 1] | [0] *
.
n = 2 case:
     |                    | b(1),b(2)
-----+--------------------+----------
   1 | [2, 2, 2, 1, 1, 1] | [0, 0]
   2 | [2, 1, 2, 1, 2, 1] | [1, 1]
   3 | [1, 2, 1, 2, 1, 2] | [1, 1]
   4 | [1, 1, 1, 2, 2, 2] | [0, 0]
.
n = 3 case:
     |                             | b(1),b(2),b(3)
-----+-----------------------------+---------------
   1 | [3, 3, 3, 2, 2, 2, 1, 1, 1] | [0, 0, 0]
   2 | [3, 3, 3, 2, 1, 2, 1, 2, 1] | [1, 1, 0]
   3 | [3, 3, 3, 1, 2, 1, 2, 1, 2] | [1, 1, 0]
   4 | [3, 3, 3, 1, 1, 1, 2, 2, 2] | [0, 0, 0]
   5 | [3, 2, 1, 3, 2, 1, 3, 2, 1] | [2, 2, 2]
   6 | [3, 1, 2, 3, 1, 2, 3, 1, 2] | [2, 2, 2]
   7 | [1, 3, 3, 3, 1, 2, 2, 2, 1] | [3, 0, 0]
   8 | [2, 3, 1, 2, 3, 1, 2, 3, 1] | [2, 2, 2]
   9 | [1, 3, 2, 1, 3, 2, 1, 3, 2] | [2, 2, 2]
  10 | [2, 1, 3, 2, 1, 3, 2, 1, 3] | [2, 2, 2]
  11 | [1, 2, 3, 1, 2, 3, 1, 2, 3] | [2, 2, 2]
  12 | [2, 2, 2, 3, 3, 3, 1, 1, 1] | [0, 0, 0]
  13 | [1, 1, 1, 3, 3, 3, 2, 2, 2] | [0, 0, 0]
  14 | [1, 2, 2, 2, 1, 3, 3, 3, 1] | [3, 0, 0]
  15 | [2, 2, 2, 1, 1, 1, 3, 3, 3] | [0, 0, 0]
  16 | [2, 1, 2, 1, 2, 1, 3, 3, 3] | [1, 1, 0]
  17 | [1, 2, 1, 2, 1, 2, 3, 3, 3] | [1, 1, 0]
  18 | [1, 1, 1, 2, 2, 2, 3, 3, 3] | [0, 0, 0]
* (strongly decreasing)

Crossrefs

Column k=3 of A332762.
Cf. A104429, A059108, A261516 (strongly decreasing), A322178, A332752.

Extensions

a(10)-a(13) from Max Alekseyev, Sep 26 2023

A104432 Number of ways to split 1, 2, 3, ..., 6n into n arithmetic progressions each with 6 terms.

Original entry on oeis.org

1, 1, 2, 4, 10, 20, 57, 119, 329, 760, 1942, 4452, 13574, 30665, 80117, 194856, 540694
Offset: 0

Views

Author

Jonas Wallgren, Mar 17 2005

Keywords

Crossrefs

Cf. A104429-A104431, A104433-A104443.

Extensions

a(0), a(11)-a(16) from Alois P. Heinz, Nov 18 2020
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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