A379633 Irregular triangle read by rows in which row n lists the largest parts of the partitions of n into consecutive parts (with the partitions ordered by increasing number of parts).

1, 2, 3, 2, 4, 5, 3, 6, 3, 7, 4, 8, 9, 5, 4, 10, 4, 11, 6, 12, 5, 13, 7, 14, 5, 15, 8, 6, 5, 16, 17, 9, 18, 7, 6, 19, 10, 20, 6, 21, 11, 8, 6, 22, 7, 23, 12, 24, 9, 25, 13, 7, 26, 8, 27, 14, 10, 7, 28, 7, 29, 15, 30, 11, 9, 8, 31, 16, 32, 33, 17, 12, 8, 34, 10, 35, 18, 9, 8, 36, 13, 8, 37, 19, 38, 11, 39, 20, 14, 9, 40, 10, 41, 21

Offset: 1

Author

Omar E. Pol, Dec 31 2024

Keywords

• nonn
• tabf

Comments

Row n gives the first A001227(n) terms of the n-th row of A379631.

Conjecture: the right border gives A212652.

Examples

			Triangle begins:
   1;
   2;
   3,  2;
   4;
   5,  3;
   6,  3;
   7,  4;
   8;
   9,  5,  4;
  10,  4;
  11,  6;
  12,  5;
  13,  7;
  14,  5;
  15,  8,  6,  5;
  16;
  17,  9;
  18,  7,  6;
  19, 10;
  20,  6;
  21, 11,  8,  6;
  22,  7;
  23, 12;
  24,  9;
  25, 13,  7;
  26,  8;
  27, 14, 10,  7;
  28,  7;
  ...
Illustration of initial terms:
                                                         _
                                                       _|1|
                                                     _|2 _|
                                                   _|3  |2|
                                                 _|4   _| |
                                               _|5    |3 _|
                                             _|6     _| |3|
                                           _|7      |4  | |
                                         _|8       _|  _| |
                                       _|9        |5  |4 _|
                                     _|10        _|   | |4|
                                   _|11         |6   _| | |
                                 _|12          _|   |5  | |
                               _|13           |7    |  _| |
                             _|14            _|    _| |5 _|
                           _|15             |8    |6  | |5|
                         _|16              _|     |   | | |
                       _|17               |9     _|  _| | |
                     _|18                _|     |7  |6  | |
                   _|19                 |10     |   |  _| |
                 _|20                  _|      _|   | |6 _|
               _|21                   |11     |8   _| | |6|
             _|22                    _|       |   |7  | | |
           _|23                     |12      _|   |   | | |
         _|24                      _|       |9    |  _| | |
       _|25                       |13       |    _| |7  | |
     _|26                        _|        _|   |8  |  _| |
   _|27                         |14       |10   |   | |7 _|
  |28                           |         |     |   | | |7|
  ...
The diagram is also the left part of the diagram of A379631.
The geometrical structure is the same as the diagram of A237591.
		

Crossrefs

Positive terms of A286013.

Column 1 gives A000027.

Row lengths give A001227.

Row sums give A286015.

Subsequence of A286000 and of A299765 and of A379631.

For the smallest parts see A379632.

Cf. A196020, A212652, A235791, A236104, A237048, A237591, A237593, A245092, A262626, A379630, A379634.