A285574 Irregular triangle read by rows which arises from a diagram which is similar to the diagram of A261699, but here the even-indexed zig-zag lines are located on the right-hand side of the vertical axis of the diagram.

1, 1, 1, 3, 1, 0, 1, 5, 1, 3, 0, 1, 0, 7, 1, 0, 0, 1, 3, 9, 1, 0, 5, 0, 1, 0, 0, 11, 1, 3, 0, 0, 1, 0, 0, 13, 1, 0, 7, 0, 1, 3, 5, 0, 15, 1, 0, 0, 0, 0, 1, 0, 0, 0, 17, 1, 3, 0, 9, 0, 1, 0, 0, 0, 19, 1, 0, 5, 0, 0, 1, 3, 0, 7, 0, 21, 1, 0, 0, 0, 11, 0, 1, 0, 0, 0, 0, 23, 1, 3, 0, 0, 0, 0, 1, 0, 5, 0, 0, 25

Offset: 1

Author

Omar E. Pol, Apr 21 2017

Keywords

• nonn
• tabf

Comments

In the diagram we have that:

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

The number of horizontal line segments in the left-hand part of the n-th row equals A082647, the number of partitions of n into an odd number of consecutive parts.

The number of horizontal line segments in the right-hand part of the n-th row equals A131576, the number of partitions of n into an even number of consecutive parts.

The diagram explains the unusual ordering of the terms in the triangle A261699, in which the even-indexed zig-zag lines are located on the left-hand side of the vertical axis of the diagram.

Examples

			Triangle begins:
1;
1;
1, 3;
1, 0,
1, 5;
1, 3, 0;
1, 0, 7;
1, 0, 0;
1, 3, 9;
1, 0, 5,  0;
1, 0, 0, 11;
1, 3, 0,  0;
1, 0, 0, 13;
1, 0, 7,  0;
1, 3, 5,  0, 15;
1, 0, 0,  0,  0;
1, 0, 0,  0, 17;
1, 3, 0,  9,  0;
1, 0, 0,  0, 19;
1, 0, 5,  0,  0;
1, 3, 0,  7,  0, 21;
...
Illustration of initial terms of the diagram:
Row                                           _
1                                           _|1|
2                                         _|1  |_
3                                       _|1    |3|
4                                     _|1      |0|_
5                                   _|1       _|  5|
6                                 _|1        |3|  0|_
7                               _|1          |0|    7|
8                             _|1           _|0|    0|_
9                           _|1            |3  |_     9|
10                        _|1              |0  |5|    0|_
11                      _|1               _|0  |0|     11|
12                    _|1                |3    |0|      0|_
13                  _|1                  |0    |0|_      13|
14                _|1                   _|0   _|  7|      0|_
15              _|1                    |3    |5|  0|       15|
16            _|1                      |0    |0|  0|        0|_
17          _|1                       _|0    |0|  0|_        17|
18        _|1                        |3      |0|    9|        0|_
19      _|1                          |0     _|0|    0|         19|
20    _|1                           _|0    |5  |_   0|          0|_
21   |1                            |3      |0  |7|  0|           21|
...
(Compare with the diagram of A261699.)
		

Crossrefs

Positive terms give A182469.

Row n has length A003056(n).

The sum of row n is A000593(n).

Column k starts in row A000217(k).

The number of positive terms in row n is A001227(n).

Cf. A082647, A131576, A196020, A235791, A236104, A237048, A237591, A237593, A245092, A259176, A259177, A261699, A279820.