OFFSET
1,2
COMMENTS
The characteristic shape of the symmetric representation of sigma(2^(n-1)) consists in that the diagram contains exactly one region (or part) and that region has width 1.
So knowing this characteristic shape we can know if a number is power of 2 or not just by looking at the diagram, even ignoring the concept of power of 2.
Therefore we can see a geometric pattern of the distribution of the powers of 2 in the stepped pyramid described in A245092.
For the definition of "width" see A249351.
T(n,k) is the length of the k-th line segment of the largest Dyck path of the symmetric representation of sigma(2^(n-1), from the border to the center, hence the sum of the n-th row of triangle is equal to A000079(n-1).
T(n,k) is also the difference between the total number of partitions of all positive integers <= 2^(n-1) into exactly k consecutive parts, and the total number of partitions of all positive integers <= 2^(n-1) into exactly k + 1 consecutive parts.
EXAMPLE
Triangle begins:
1;
2;
3, 1;
5, 2, 1;
9, 3, 2, 1, 1;
17, 6, 3, 2, 2, 1, 1;
33, 11, 6, 4, 2, 2, 2, 1, 2, 1;
65, 22, 11, 7, 5, 3, 3, 2, 2, 2, 1, 2, 1, 1, 1;
129, 43, 22, 13, 9, 7, 5, 4, 3, 3, 3, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1;
...
Illustration of initial terms:
.
Row 1: _
|_| Semilength = 1
1
Row 2: _
_| |
|_ _|
2 Semilength = 2
.
Row 3: _
| |
_| |
_ _| _|
|_ _ _|1 Semilength = 4
3
.
Row 4: _
| |
| |
| |
_ _| |
_| _ _|
| _|
_ _ _ _| | 1 Semilength = 8
|_ _ _ _ _|2
5
.
Row 5: _
| |
| |
| |
| |
| |
| |
| |
_ _ _| |
| _ _ _|
_| |
_| _|
_ _| _| Semilength = 16
| _ _|1 1
| | 2
_ _ _ _ _ _ _ _| |3
|_ _ _ _ _ _ _ _ _|
9
.
The area (also the number of cells) of the successive diagrams gives the nonzero Mersenne numbers A000225.
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Aug 06 2021
STATUS
approved