OFFSET
1,2
COMMENTS
The Mersenne number A000225(n) does not has a characteristic shape of its symmetric representation of sigma(A000225(n)). On the other hand, we can find that number in two ways in the symmetric representation of the powers of 2 as follows: the Mersenne numbers are the semilength of the smallest Dyck path and also they equals the area (or the number of cells) of the region of the diagram (see examples).
Therefore we can see a geometric pattern of the distribution of the Mersenne numbers in the stepped pyramid described in A245092.
EXAMPLE
Triangle begins:
1;
2, 1;
4, 2, 1;
8, 3, 2, 1, 1;
16, 6, 3, 2, 2, 1, 1;
32, 11, 6, 4, 2, 2, 2, 1, 2, 1;
64, 22, 11, 7, 5, 3, 3, 2, 2, 2, 1, 2, 1, 1, 1;
128, 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:
0_ Semilength = 0 Area = 1
|_|
Row 2:
_
1_| | Semilength = 1 Area = 3
|_ _|
.
Row 3: _
| |
1 _| |
2_ _| _| Semilength = 3 Area = 7
|_ _ _|
.
Row 4: _
| |
| |
| |
_ _| |
1 _| _ _|
4 2| _| Semilength = 7 Area = 15
_ _ _ _| |
|_ _ _ _ _|
.
Row 5: _
| |
| |
| |
| |
| |
| |
| |
_ _ _| |
| _ _ _|
_| |
1 1_| _|
2 _ _| _| Semilength = 15 Area = 31
| _ _|
8 3| |
_ _ _ _ _ _ _ _| |
|_ _ _ _ _ _ _ _ _|
.
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Aug 06 2021
STATUS
approved