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 1-4 of 4 results.

A278339 Number of set partitions of [n] into subsets whose element sums are distinct squares.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 1, 0, 1, 23, 9, 41, 248, 277, 1556, 2854, 5233, 20701, 145137, 1626890, 4118910, 9963276, 9260756, 64027363, 365237571, 1002679107, 21594036300, 24465529531, 144914973347, 1921444799766
Offset: 0

Views

Author

Alois P. Heinz, Nov 18 2016

Keywords

Examples

			a(0) = 1: {}.
a(1) = 1: 1.
a(4) = 1: 1|234.
a(6) = 1: 1|2356|4.
a(8) = 1: 12345678.
a(9) = 23: 12345678|9, 123568|4|79, 1236789|45, 1245789|36, 126|345789, 12589|367|4, 1258|3679|4, 12679|358|4, 1267|3589|4, 1345689|27, 135|246789, 13|24568|79, 13579|268|4, 1357|2689|4, 13678|259|4, 13|259|4678, 13|2689|457, 13|268|4579, 156789|234, 18|2345679, 169|23578|4, 1789|2356|4, 178|23569|4.
a(10) = 9: 1|2356|4|78(10)|9, 1|23578|4|6(10)|9, 1|258(10)|367|4|9, 1|258(10)|36|4|79, 1|259|36|4|78(10), 1|267(10)|358|4|9, 1|268|357(10)|4|9, 1|27|3589|4|6(10), 1|27|358|4|69(10).

Crossrefs

Cf. A252897, A275780, A278329, A278340, A281994.

A278329 Number of set partitions of [3n] into n subsets of size three such that all element sums are squares.

Original entry on oeis.org

1, 0, 0, 0, 0, 29, 35, 340, 26579, 390480, 9514434, 145963193, 5474045270, 87251356528, 5454606723223, 182600931998737, 5059541554893941
Offset: 0

Views

Author

Alois P. Heinz, Nov 18 2016

Keywords

Examples

			a(5) = 29: {{1,3,5}, {2,9,14}, {4,6,15}, {7,8,10}, {11,12,13}},
  {{1,3,5}, {2,11,12}, {4,6,15}, {7,8,10}, {9,13,14}},
  {{1,2,6}, {3,7,15}, {4,8,13}, {5,9,11}, {10,12,14}},
  {{1,2,6}, {3,7,15}, {4,10,11}, {5,8,12}, {9,13,14}},
  {{1,3,5}, {2,8,15}, {4,7,14}, {6,9,10}, {11,12,13}},
  {{1,3,5}, {2,8,15}, {4,10,11}, {6,7,12}, {9,13,14}},
  {{1,9,15}, {2,3,4}, {5,6,14}, {7,8,10}, {11,12,13}},
  {{1,9,15}, {2,3,4}, {5,7,13}, {6,8,11}, {10,12,14}},
  {{1,2,6}, {3,9,13}, {4,7,14}, {5,8,12}, {10,11,15}},
  {{1,2,6}, {3,8,14}, {4,9,12}, {5,7,13}, {10,11,15}},
  {{1,3,5}, {2,9,14}, {4,8,13}, {6,7,12}, {10,11,15}},
  {{1,3,12}, {2,6,8}, {4,5,7}, {9,13,14}, {10,11,15}},
  {{1,11,13}, {2,3,4}, {5,6,14}, {7,8,10}, {9,12,15}},
  {{1,3,5}, {2,10,13}, {4,7,14}, {6,8,11}, {9,12,15}},
  {{1,2,6}, {3,8,14}, {4,10,11}, {5,7,13}, {9,12,15}},
  {{1,10,14}, {2,3,4}, {5,7,13}, {6,8,11}, {9,12,15}},
  {{1,2,6}, {3,10,12}, {4,7,14}, {5,9,11}, {8,13,15}},
  {{1,3,5}, {2,11,12}, {4,7,14}, {6,9,10}, {8,13,15}},
  {{1,3,5}, {2,9,14}, {4,10,11}, {6,7,12}, {8,13,15}},
  {{1,10,14}, {2,3,4}, {5,9,11}, {6,7,12}, {8,13,15}},
  {{1,6,9}, {2,3,11}, {4,5,7}, {8,13,15}, {10,12,14}},
  {{1,4,11}, {2,5,9}, {3,6,7}, {8,13,15}, {10,12,14}},
  {{1,2,6}, {3,10,12}, {4,8,13}, {5,9,11}, {7,14,15}},
  {{1,3,5}, {2,11,12}, {4,8,13}, {6,9,10}, {7,14,15}},
  {{1,2,6}, {3,9,13}, {4,10,11}, {5,8,12}, {7,14,15}},
  {{1,3,5}, {2,10,13}, {4,9,12}, {6,8,11}, {7,14,15}},
  {{1,11,13}, {2,3,4}, {5,8,12}, {6,9,10}, {7,14,15}},
  {{1,6,9}, {2,4,10}, {3,5,8}, {7,14,15}, {11,12,13}},
  {{1,5,10}, {2,6,8}, {3,4,9}, {7,14,15}, {11,12,13}}.

Crossrefs

Cf. A000290, A252897, A278339, A281706, A295946.

Programs

  • Mathematica
    A278329[0] = 1;
A278329[n_] := Length@FindClique[Graph[First@# <-> Last@# & /@ Select[Subsets[Select[Flatten[IntegerPartitions[#^2, {3}, Range[3 n]] & /@ Range[Sqrt[9 n - 3]], 1], DuplicateFreeQ], {2}], DuplicateFreeQ@Flatten@# &]], {n}, All] (* Davin Park, Jan 26 2017 *)

Extensions

a(12)-a(13) from Bert Dobbelaere, Apr 12 2019
a(14)-a(16) from Martin Fuller, Jul 08 2025

A253472 Square Pairs: Numbers n such that 1, 2, ..., 2n can be partitioned into n pairs, where each pair adds up to a perfect square.

Original entry on oeis.org

4, 7, 8, 9, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80
Offset: 1

Views

Author

Henri Picciotto, Jan 01 2015

Keywords

Comments

Kiran Kedlaya proved that all numbers greater than 11 are included in the sequence. Outline of proof:
- Show by hand or by computer that it works up to n = 30.
- For n=31, pair 62+59=61+60=11^2 and then reduce to the case of n=29. For n=32, pair 64+57, ..., 61+60 and reduce to the case of 28. And so on. This works until n=48.
- For n=49, ..., 72 pairs adding up to 13^2 allow us to reduce to n=35.
- Repeat the process, always terminating at (2m+1)^2-25, aiming for sums of (2m+3)^2. The first such pair is (2m+1)^2-23, 8m+31.
- This always works, as long as (2m+1)^2 - 25 > 8m+31, and therefore we must have m > 4.
A similar sequence using odd numbers can be created, by making n pairs that sum to perfect squares, using numbers from 0 to 2n-1. All numbers greater than 6 are included.
Worthy of consideration for the elementary school classroom working on square numbers. - Gordon Hamilton, Mar 20 2015

Examples

			For n = 4: (8, 1), (7, 2), (6, 3), (5, 4).
For n = 7: (14, 2), (13, 3), (12, 4), (11, 5), (10, 6), (9, 7), (8, 1).

References

  • Alfred S. Posamentier, Stephen Krulik, Problem-Solving Strategies for Efficient and Elegant Solutions, Grades 6-12, 2008, page 191.

Links

Crossrefs

Cf. A252897.

Programs

  • Python
    # See link.

Formula

From Chai Wah Wu, Aug 13 2020: (Start)
a(n) = 2*a(n-1) - a(n-2) for n > 6.
G.f.: x*(-2*x^5 + 2*x^4 - 2*x^2 - x + 4)/(x - 1)^2. (End)

A257542 Square-sum pairs: Numbers n such that 0,1, ..., 2n-1 can be partitioned into n pairs, where each pair adds up to a perfect square.

Original entry on oeis.org

1, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75
Offset: 1

Views

Author

Brian Hopkins, Apr 28 2015

Keywords

Comments

Kilkelly uses induction to prove that all integers greater than 20 are in the sequence after using various methods on smaller cases.
The positive integers except 2, 3, and 6.
The positive integers except the strong divisors of 6. - Omar E. Pol, Apr 30 2015

Examples

			For n = 4: (0, 1), (2, 7), (3, 6), (4, 5)
For n = 7: (0, 9), (1, 8), (2, 7), (3, 13), (4, 12), (5, 11), (6, 10)

References

  • T. Kilkelly, The ARML Power Contest, American Mathematical Society, 2015, chapter 11.

Links

Crossrefs

Cf. A253472, A252897.
Essentially the same as A055495.

Programs

Formula

From Chai Wah Wu, Aug 13 2020: (Start)
a(n) = 2*a(n-1) - a(n-2) for n > 5.
G.f.: x*(-x^4 + x^3 - 2*x^2 + 2*x + 1)/(x - 1)^2. (End)
Showing 1-4 of 4 results.

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