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A346872
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Irregular triangle read by rows in which row n lists the row 2^(n-1) of A237591, n >= 1.
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7
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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, 257, 86, 43, 26, 18, 12, 10, 8, 6, 5, 4, 4, 3, 3, 3, 2, 3
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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:
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Row 1: _
|_| Semilength = 1
1
Row 2: _
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2 Semilength = 2
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Row 3: _
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_ _| _|
|_ _ _|1 Semilength = 4
3
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Row 4: _
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_ _| |
_| _ _|
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_ _ _ _| | 1 Semilength = 8
|_ _ _ _ _|2
5
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Row 5: _
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_ _ _| |
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_ _| _| Semilength = 16
| _ _|1 1
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_ _ _ _ _ _ _ _| |3
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9
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The area (also the number of cells) of the successive diagrams gives the nonzero Mersenne numbers A000225.
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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