OFFSET
1,2
COMMENTS
EXAMPLE
Triangle begins:
[1];
[2];
[3],
[4];
[5];
[6], [3, 2, 1];
[7];
[8];
[9], [4, 3, 2];
[10];
[11];
[12], [5, 4, 3];
[13];
[14];
[15], [6, 5, 4], [5, 4, 3, 2, 1];
[16];
[17];
[18], [7, 6, 5];
[19];
[20], [6, 5, 4, 3, 2];
[21], [8, 7, 6];
[22];
[23];
[24], [9, 8, 7];
[25], [7, 6, 5, 4, 3];
[26];
[27], [10, 9, 8];
[28], [7, 6, 5, 4, 3, 2, 1];
...
In the diagram below the m-th staircase walk starts at row A000384(m).
The number of horizontal line segments in the n-th row equals A082647(n), the number of partitions of n into an odd number of consecutive parts, so we can find such partitions as follows: consider the vertical blocks of numbers that start exactly in the n-th row of the diagram, for example: for n = 15 consider the vertical blocks of numbers that start exactly in the 15th row. They are [15], [6, 5, 4]. [5, 4, 3, 2, 1], equaling the 15th row of the above triangle.
_
_|1|
_|2 |
_|3 |
_|4 |
_|5 _|
_|6 |3|
_|7 |2|
_|8 _|1|
_|9 |4 |
_|10 |3 |
_|11 _|2 |
_|12 |5 |
_|13 |4 |
_|14 _|3 _|
_|15 |6 |5|
_|16 |5 |4|
_|17 _|4 |3|
_|18 |7 |2|
_|19 |6 _|1|
_|20 _|5 |6 |
_|21 |8 |5 |
_|22 |7 |4 |
_|23 _|6 |3 |
_|24 |9 _|2 |
_|25 |8 |7 |
_|26 _|7 |6 |
_|27 |10 |5 _|
|28 |9 |4 |7|
...
The diagram is infinite.
For more information about the diagram see A286000.
CROSSREFS
Subsequence of A299765.
Row sums give A352257.
Column 1 gives A000027.
Records give A000027.
Row n contains A082647(n) of the mentioned partitions.
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Mar 15 2022
STATUS
approved