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 1-9 of 9 results.

A279199 Number of reducible ways to split 1, 2, 3, ..., 3n into n arithmetic progressions each with 3 terms: a(n) = A104429(n) - A202705(n).

Original entry on oeis.org

0, 0, 1, 3, 9, 30, 117, 512, 2597, 14892, 99034, 721350, 5909324, 52578654, 516148082, 5422071091, 61889692290, 749456672155, 9767058240577, 134007989313530, 1958535749524107
Offset: 0

Views

Author

N. J. A. Sloane, Dec 15 2016

Keywords

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.

Links

  • R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [Annotated scanned copy, with permission] See sequence "L".
  • 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

a(n) = A104429(n)-A202705(n) = Sum_{i=1..n-1} A104429(i)*A202705(n-i). - Martin Fuller, Jul 08 2025

Extensions

Definition corrected by N. J. A. Sloane, Jan 09 2017 at the suggestion of Fausto A. C. Cariboni.
a(15)-a(17) from Fausto A. C. Cariboni, Feb 22 2017
a(18)-a(20) from Martin Fuller, Jul 08 2025

A104429 Number of ways to split {1, 2, 3, ..., 3n} into n arithmetic progressions each with 3 terms.

Original entry on oeis.org

1, 1, 2, 5, 15, 55, 232, 1161, 6643, 44566, 327064, 2709050, 24312028, 240833770, 2546215687, 29251369570, 355838858402, 4658866773664, 64127566159756, 940320691236206
Offset: 0

Views

Author

Jonas Wallgren, Mar 17 2005

Keywords

Examples

			{{{1,2,3},{4,5,6},{7,8,9}}, {{1,2,3},{4,6,8},{5,7,9}}, {{1,3,5},{2,4,6},{7,8,9}}, {{1,4,7},{2,5,8},{3,6,9}}, {{1,5,9},{2,3,4},{6,7,8}}} are the 5 ways to split 1, 2, 3, ..., 9 into 3 arithmetic progressions each with 3 elements. Thus a(3)=5.

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.

Links

Crossrefs

Cf. A104430-A104443.
All of A279197, A279198, A202705, A279199, A282615 are concerned with counting solutions to X+Y=2Z in various ways.
See also A002848, A002849, A334250.

Extensions

a(11)-a(14) from Alois P. Heinz, Dec 28 2011
a(15)-a(17) from Fausto A. C. Cariboni, Feb 22 2017
a(18)-a(19) from Martin Fuller, Jul 08 2025

A279197 Number of self-conjugate inseparable solutions of X + Y = 2Z (integer, disjoint triples from {1,2,3,...,3n}).

Original entry on oeis.org

1, 1, 2, 2, 11, 11, 55, 58, 486, 442, 4218, 3924, 45096, 42013, 538537, 505830, 7368091, 6959545, 111877294, 105723374, 1886636688, 1763443165, 34585786729, 32401780965, 687085545694, 642233156868, 14691047314846, 13788837896728, 340221989868538, 317342350394678, 8327884506579315
Offset: 1

Views

Author

N. J. A. Sloane, Dec 15 2016

Keywords

Comments

In Richard Guy's letter, the term 50 is marked with a question mark. Peter Kagey has shown that the value should be 55. - N. J. A. Sloane, Feb 15 2017
From Peter Kagey, Feb 14 2017: (Start)
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.
(End)

Examples

			Examples of solutions X,Y,Z for n=5:
  2,4,3
  5,7,6
  1,15,8
  9,11,10
  12,14,13
and in his letter Richard Guy has drawn links pairing the first and fifth solutions, and the second and fourth solutions.
For n = 2 the a(2) = 1 solution is
  [(2,6,4),(1,5,3)].
For n = 3 the a(3) = 2 solutions are
  [(1,7,4),(3,9,6),(2,8,5)] and
  [(2,4,3),(6,8,7),(1,9,5)].

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.

Links

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

a(n) = A282616(n) - A282615(n). - Martin Fuller, Jul 15 2025

Extensions

a(7) corrected and a(8)-a(13) added by Peter Kagey, Feb 14 2017
a(14)-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

A282615 Number of self-conjugate separable solutions of X + Y = 2Z (integer, disjoint triples from {1,2,3,...,3n}).

Original entry on oeis.org

0, 1, 1, 3, 4, 9, 20, 35, 102, 160, 736, 930, 5972, 6766, 59017, 61814, 671651, 675379, 8844028, 8675583, 130880467, 126385830, 2163551657, 2049560059, 39112954305, 36883483406, 768337929193, 720918897940, 16279025598443, 15303083773040, 373743187469167, 349148771223261, 9095126347788632
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) = 3 solutions are:
  (10,12,11),(7,9,8),(4,6,5),(1,3,2),
  (10,12,11),(5,9,7),(4,8,6),(1,3,2), and
  (8,12,10),(7,11,9),(2,6,4),(1,5,3).

Crossrefs

Cf. A104429, A202705, A279197, A279199, A282616, A282617, A282618, A282619.
All of A279197, A279198, A202705, A279199, A104429, A282615 are concerned with counting solutions to X+Y=2Z in various ways.

Formula

a(n) = A282616(n) - A279197(n).
a(n) = A279199(n) - A282618(n).
a(n) = Sum_{i=1..floor(n/2)} A202705(i) * (A282616(n-2*i) if n>2*i else 1) = Sum_{i=1..floor(n/2)} A104429(i) * (A279197(n-2*i) if n>2*i else 1). - Martin Fuller, Jul 15 2025

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(33) from Martin Fuller, Jul 15 2025

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

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
Showing 1-9 of 9 results.

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