login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A286001 A table of partitions into consecutive parts (see Comments lines for definition). 43
1, 2, 3, 1, 4, 2, 5, 2, 6, 3, 1, 7, 3, 2, 8, 4, 3, 9, 4, 2, 10, 5, 3, 1, 11, 5, 4, 2, 12, 6, 3, 3, 13, 6, 4, 4, 14, 7, 5, 2, 15, 7, 4, 3, 1, 16, 8, 5, 4, 2, 17, 8, 6, 5, 3, 18, 9, 5, 3, 4, 19, 9, 6, 4, 5, 20, 10, 7, 5, 2, 21, 10, 6, 6, 3, 1, 22, 11, 7, 4, 4, 2, 23, 11, 8, 5, 5, 3, 24, 12, 7, 6, 6, 4, 25, 12, 8, 7, 3, 5 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

This is a triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists successive blocks of k consecutive terms, where the m-th block starts with m, m>=1, and the first element of column k is in row k*(k+1)/2.

The partitions of n into consecutive parts are represented from the row n up to row A288529(n) as maximum, but in increasing order, exclusively in the columns where the blocks begin.

More precisely, the partition of n into exactly k consecutive parts (if such partition exists) is represented in the column k from the row n up to row n + k - 1 (see examples).

A288772(n) is the minimum number of rows that are required to represent in this table the partitions of all positive integers <= n into consecutive parts.

A288773(n) is the largest of all positive integers whose partitions into consecutive parts can be totally represented in the first n rows of this table.

A288774(n) is the largest positive integers whose partitions into consecutive parts can be totally represented in the first n rows of this table.

For a theorem related to this table see A286000.

LINKS

Table of n, a(n) for n=1..100.

EXAMPLE

Triangle begins:

1;

2;

3,   1;

4,   2;

5,   2;

6,   3,  1;

7,   3,  2;

8,   4,  3;

9,   4,  2;

10,  5,  3,  1;

11,  5,  4,  2;

12,  6,  3,  3;

13,  6,  4,  4;

14,  7,  5,  2;

15,  7,  4,  3,  1;

16,  8,  5,  4,  2;

17,  8,  6,  5,  3;

18,  9,  5,  3,  4;

19,  9,  6,  4,  5;

20, 10,  7,  5,  2;

21, 10,  6,  6,  3,  1;

22, 11,  7,  4,  4,  2;

23, 11,  8,  5,  5,  3;

24, 12,  7,  6,  6,  4;

25, 12,  8,  7,  3,  5;

26, 13,  9,  5,  4,  6;

27, 13,  8,  6,  5,  2;

28, 14,  9,  7,  6,  3,  1;

...

Figures A..G show the location (in the columns of the table) of the partitions of n = 1..7 (respectively) into consecutive parts:

.   ------------------------------------------------------------------------

Fig:   A     B       C         D          E            F             G

.   ------------------------------------------------------------------------

. n:   1     2       3         4          5            6             7

Row ------------------------------------------------------------------------

1   | [1];|  1; |  1;     |  1;    |  1;        |  1;         |  1;        |

2   |     | [2];|  2;     |  2;    |  2;        |  2;         |  2;        |

3   |     |     | [3],[1];|  3,  1;|  3,  1;    |  3,  1;     |  3,  1;    |

4   |     |     |  4 ,[2];| [4], 2;|  4,  2;    |  4,  2;     |  4,  2;    |

5   |     |     |         |        | [5],[2];   |  5,  2;     |  5,  2;    |

6   |     |     |         |        |  6, [3], 3;| [6], 3, [1];|  6,  3,  1;|

7   |     |     |         |        |            |  7,  3, [2];| [7],[3], 2;|

8   |     |     |         |        |            |  8,  4, [3];|  8, [4], 3;|

.   ------------------------------------------------------------------------

Figure F: for n = 6 the partitions of 6 into consecutive parts (but with the parts in increasing order) are [6] and [1, 2, 3]. These partitions have 1 and 3 consecutive parts respectively. On the other hand  we can find the mentioned partitions in the columns 1 and 3 of this table, starting at the row 6.

.

Figures H..K show the location (in the columns of the table) of the partitions of 8..11 (respectively) into consecutive parts:

.    --------------------------------------------------------------------

Fig:        H             I                  J                 K

.    --------------------------------------------------------------------

. n:        8             9                  10                11

Row  --------------------------------------------------------------------

1    |  1;        |  1;            |   1;             |   1;            |

1    |  2;        |  2;            |   2;             |   2;            |

3    |  3,  1;    |  3,  1;        |   3,  1;         |   3,  1;        |

4    |  4,  2;    |  4,  2;        |   4,  2;         |   4,  2;        |

5    |  5,  2;    |  5,  2;        |   5,  2;         |   5,  2;        |

6    |  6,  3,  3;|  6,  3,  1;    |   6,  3,  1;     |   6,  3,  1;    |

7    |  7,  3,  2;|  7,  3,  2;    |   7,  3,  2;     |   7,  3,  2;    |

8    | [8], 4,  1;|  8,  4,  3;    |   8,  4,  3;     |   8,  4,  3;    |

9    |            | [9],[4],[2];   |   9,  4,  2;     |   9,  4,  2;    |

10   |            | 10, [5],[3], 1;| [10], 5,  3, [1];|  10,  5,  3,  1;|

11   |            | 11,  5, [4], 2;|  11,  5,  4, [2];| [11],[5], 4,  2;|

12   |            |                |  12,  6,  3, [3];|  12, [6], 3,  3;|

13   |            |                |  13,  6,  4, [4];|  13,  6,  4,  4;|

.    --------------------------------------------------------------------

Figure J: For n = 10 the partitions of 10 into consecutive parts (but with the parts in increasing order) are [10] and [1, 2, 3, 4]. These partitions have 1 and 4 consecutive parts respectively. On the other hand, we can find the mentioned partitions in the columns 1 and 4 of this table, starting at the row 10.

.

Illustration of initial terms arranged into the diagram of the triangle A237591:

.                                                           _

.                                                         _|1|

.                                                       _|2 _|

.                                                     _|3  |1|

.                                                   _|4   _|2|

.                                                 _|5    |2 _|

.                                               _|6     _|3|1|

.                                             _|7      |3  |2|

.                                           _|8       _|4 _|3|

.                                         _|9        |4  |2 _|

.                                       _|10        _|5  |3|1|

.                                     _|11         |5   _|4|2|

.                                   _|12          _|6  |3  |3|

.                                 _|13           |6    |4 _|4|

.                               _|14            _|7   _|5|2 _|

.                             _|15             |7    |4  |3|1|

.                           _|16              _|8    |5  |4|2|

.                         _|17               |8     _|6 _|5|3|

.                       _|18                _|9    |5  |3  |4|

.                     _|19                 |9      |6  |4 _|5|

.                   _|20                  _|10    _|7  |5|2 _|

.                 _|21                   |10     |6   _|6|3|1|

.               _|22                    _|11     |7  |4  |4|2|

.             _|23                     |11      _|8  |5  |5|3|

.           _|24                      _|12     |7    |6 _|6|4|

.         _|25                       |12       |8   _|7|3  |5|

.       _|26                        _|13      _|9  |5  |4 _|6|

.     _|27                         |13       |8    |6  |5|2 _|

.    |28                           |14       |9    |7  |6|3|1|

...

The number of horizontal line segments in the n-th row of the diagram equals A001227(n), the number of partitions of n into consecutive parts.

.

From Omar E. Pol, Dec 15 2020: (Start)

The connection (described step by step) between the triangle of A299765 and the above geometric diagram is as follows:

.

   [1];                                       [1];

   [2];                                       [2];

   [3], [2, 1];                               [3], [2, 1];

   [4];                                       [4];

   [5], [3, 2];                               [5], [3, 2];

   [6], [3, 2, 1];                            [6],         [3, 2, 1];

   [7], [4, 3];                               [7], [4, 3];

   [8];                                       [8];

   [9], [5, 4], [4, 3, 2];                    [9], [5, 4], [4, 3, 2];

.

         Figure 1.                                   Figure 2.

.

We start with the irregular                Then we write the same triangle

triangle of A299765 in which               but ordered in columns where the

row n lists the partitions                 column k lists the partitions of

of n into consecutive parts.               n into k consecutive parts.

.

.   _                                          _

    1|                                        |1

    _                                          _

    2|                                        |2

    _    _ _                                   _      _

    3|   2,1|                                 |3     |1

    _                                          _     |2

    4|                                        |4

    _    _ _                                   _      _

    5|   3,2|                                 |5     |2

    _           _ _ _                          _     |3      _

    6|          3,2,1|                        |6            |1

    _    _ _                                   _      _     |2

    7|   4,3|                                 |7     |3     |3

    _                                          _     |4

    8|                                        |8

    _    _ _    _ _ _                          _      _      _

    9|   5,4|   4,3,2|                        |9     |4     |2

                                                     |5     |3

                                                            |4

.

         Figure 3.                                Figure 4.

.

Then we draw to the right of               The we rotate each sub-diagram

each partition a vertical                  90 degrees counterclockwise.

toothpick and above each part              Every horizontal toothpick represents

we draw a horizontal toothpick.            the existence of that partition.

.                                          The number of vertical toothpicks

.                                          equals the number of parts.

.

.                     _                                      _

                    _|1                                    _|1

                  _|2 _                                  _|2 _

                _|3  |1                                _|3  |1

              _|4   _|2                              _|4   _|2

            _|5    |2 _                            _|5    |2 _

          _|6     _|3|1                          _|6     _|3|1

        _|7      |3  |2                        _|7      |3  |2

      _|8       _|4 _|3                      _|8       _|4 _|3

     |9        |4  |2                       |9        |4  |2

               |5  |3

                   |4

.

         Figure 5.                                Figure 6.

.

Then we join the sub-diagrams              Finally we erase the parts that

forming staircases (or zig-zag             are beyond a certain level (in

paths) that represent the                  this case beyond the 9th level)

partitions that have the same              to make the diagram more standard.

number of parts.

.

The numbers in the k-th staircase (from left to right) are the elements of the k-th column of the triangular array.

Note that this diagram is essentially the same diagram used to represent the triangles A237048, A235791, A237591, and other related sequences such as A001227, A060831 and A204217.

There is an infinite family of this kind of triangles, which are related to polygonal numbers and partitions into consecutive parts that differ by d. For more information see the theorems in A285914 and A303300.

Note that if we take two images of the diagram mirroring each other, with the y-axis in the middle of them, then a new diagram is formed, which is symmetric and represents the sequence A237593 as an isosceles triangle. Then if we fold each level (or row) of that isosceles triangle we essentially obtain the structure of the pyramid described in A245092 whose terraces at the n-th level have a total area equal to sigma(n) = A000203(n). (End)

CROSSREFS

Another version of A286000.

Tables of the same family where the consecutive parts differ by d are A010766 (d=0), this sequence (d=1), A332266 (d=2), A334945 (d=3), A334618(d=4).

Cf. A000217, A001227, A003056, A109814, A196020, A204217, A211343, A235791, A236104, A237048, A237591, A237593, A245092, A262626, A280850, A280851, A285914, A286013, A288529, A288772, A288773, A288774, A296508, A299765, A303300.

Sequence in context: A152547 A083906 A160541 * A304106 A022446 A122196

Adjacent sequences:  A285998 A285999 A286000 * A286002 A286003 A286004

KEYWORD

nonn,tabl

AUTHOR

Omar E. Pol, Apr 30 2017

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 16 04:49 EDT 2021. Contains 343030 sequences. (Running on oeis4.)