A332266 Irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which column k lists successive blocks of k consecutive integers that differ by 2, where the m-th block starts with m, m >= 1, and the first element of column k is in row k^2.

1, 2, 3, 4, 1, 5, 3, 6, 2, 7, 4, 8, 3, 9, 5, 1, 10, 4, 3, 11, 6, 5, 12, 5, 2, 13, 7, 4, 14, 6, 6, 15, 8, 3, 16, 7, 5, 1, 17, 9, 7, 3, 18, 8, 4, 5, 19, 10, 6, 7, 20, 9, 8, 2, 21, 11, 5, 4, 22, 10, 7, 6, 23, 12, 9, 8, 24, 11, 6, 3, 25, 13, 8, 5, 1, 26, 12, 10, 7, 3, 27, 14, 7, 9, 5

Offset

1,2

Comments

This triangle can be interpreted as a table of partitions into consecutive parts that differ by 2 (see the Example section).

Links

Table of n, a(n) for n=1..85.

Example

Triangle begins:
   1;
   2;
   3;
   4,  1;
   5,  3;
   6,  2;
   7,  4;
   8,  3;
   9,  5,  1;
  10,  4,  3;
  11,  6,  5;
  12,  5,  2;
  13,  7,  4;
  14,  6,  6;
  15,  8,  3;
  16,  7,  5,  1;
  17,  9,  7,  3;
  18,  8,  4,  5;
  19, 10,  6,  7;
  20,  9,  8,  2;
  21, 11,  5,  4;
  22, 10,  7,  6;
  23, 12,  9,  8;
  24, 11,  6,  3;
  25, 13,  8,  5,  1;
...
Figures A..G show the location (in the columns of the table) of the partitions of n = 1..7 (respectively) into consecutive parts that differ by 2:
.   ---------------------------------------------------------
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];|  3;     |  3;    |  3;     |  3;    |
4   |     |     |     | [4],[1];|  4,  1;|  4,  1; |  4,  1;|
5   |     |     |     |  5, [3];| [5], 3;|  5,  3; |  5,  3;|
6   |     |     |     |         |        | [6],[2];|  6,  2;|
7   |     |     |     |         |        |  7, [4];| [7], 4;|
.   ---------------------------------------------------------
Figure F: for n = 6 the partitions of 6 into consecutive parts that differ by 2 (but with the parts in increasing order) are [6] and [2, 4]. These partitions have one part and two parts respectively. On the other hand we can find the mentioned partitions in the columns 1 and 2 of this table, starting at the row 6.
.
Figures H..L show the location (in the columns of the table) of the partitions of 8..12 (respectively) into consecutive parts that differ by 2:
.    -----------------------------------------------------------------------
Fig:        H            I            J              K             L
.    -----------------------------------------------------------------------
. n:        8            9            10             11            12
Row  -----------------------------------------------------------------------
1    |  1;        |  1;         |   1;        |   1;        |   1;         |
1    |  2;        |  2;         |   2;        |   2;        |   2;         |
3    |  3;        |  3;         |   3;        |   3;        |   3;         |
4    |  4,  1;    |  4,  1;     |   4,  1;    |   4,  1;    |   4,  1;     |
5    |  5,  3;    |  5,  3;     |   5,  3;    |   5,  3;    |   5,  3;     |
6    |  6,  2;    |  6,  2;     |   6,  2;    |   6,  2;    |   6,  2;     |
7    |  7,  4;    |  7,  4;     |   7,  4;    |   7,  4;    |   7,  4;     |
8    | [8],[3];   |  8,  3;     |   8,  3;    |   8,  3;    |   8,  3;     |
9    |  9, [5], 1;| [9], 5, [1];|   9,  5,  1;|   9,  5,  1;|   9,  5,  1; |
10   |            | 10,  4, [3];| [10],[4], 3;|  10,  4,  3;|  10,  4;  3; |
11   |            | 11,  6, [5];|  11, [6], 5;| [11], 6,  5,|  11,  6;  5; |
12   |            |             |             |             | [12],[5],[2];|
13   |            |             |             |             |  13, [7],[4];|
14   |            |             |             |             |  14,  6, [6];|
.    -----------------------------------------------------------------------
Figure I: for n = 9 the partitions of 9 into consecutive parts that differ by 2(but with the parts in increasing order) are [9] and [1, 3, 5]. These partitions have one part and three 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 9.
.
Illustration of initial terms arranged into a triangular structure:
.                                                           _
.                                                         _|1|
.                                                       _|2  |
.                                                     _|3   _|
.                                                   _|4    |1|
.                                                 _|5     _|3|
.                                               _|6      |2  |
.                                             _|7       _|4  |
.                                           _|8        |3   _|
.                                         _|9         _|5  |1|
.                                       _|10         |4    |3|
.                                     _|11          _|6   _|5|
.                                   _|12           |5    |2  |
.                                 _|13            _|7    |4  |
.                               _|14             |6     _|6  |
.                             _|15              _|8    |3   _|
.                           _|16               |7      |5  |1|
.                         _|17                _|9     _|7  |3|
.                       _|18                 |8      |4    |5|
.                     _|19                  _|10     |6   _|7|
.                   _|20                   |9       _|8  |2  |
.                 _|21                    _|11     |5    |4  |
.               _|22                     |10       |7    |6  |
.             _|23                      _|12      _|9   _|8  |
.           _|24                       |11       |6    |3   _|
.          |25                         |13       |8    |5  |1|
...
The number of horizontal line segments in the n-th row of the diagram equals A038548(n), the number of partitions of n into consecutive parts that differ by 2.

Crossrefs

Tables of the same family where the consecutive parts differ by d are A010766 (d=0), A286001 (d=1), this sequence (d=2), A334945 (d=3), A334618(d=4).

Cf. A038548, A060872, A066839, A303300, A330466.

Sequence in context: A276951 A071437 A243713 * A129709 A253146 A253028

Adjacent sequences: A332263 A332264 A332265 * A332267 A332268 A332269

Keyword

nonn,tabf

Author

Omar E. Pol, Feb 08 2020

Status

approved