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

1,2

Comments

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

Conjecture: the right border gives A212652.

Links

Table of n, a(n) for n=1..94.

Example

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.

Sequence in context: A172520 A387188 A290094 * A285904 A285714 A161908

Adjacent sequences: A379630 A379631 A379632 * A379634 A379635 A379636

Keyword

nonn,tabf

Author

Omar E. Pol, Dec 31 2024

Status

approved