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EXAMPLE
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Triangle begins:
1;
1, 2, 1;
1, 2, 0, 2, 1;
1, 2, 3, 4, 3, 2, 1;
1, 2, 3, 0, 0, 0, 3, 2, 1;
1, 2, 3, 4, 5, 7, 5, 4, 3, 2, 1;
1, 2, 3, 4, 0, 0, 0, 0, 0, 4, 3, 2, 1;
1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 2, 1;
1, 2, 3, 4, 5, 0, 0, 1, 4, 1, 0, 0, 5, 4, 3, 2, 1;
1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 9, 8, 7, 6, 5, 4, 3, 2, 1;
1, 2, 3, 4, 5, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 5, 4, 3, 2, 1;
...
Written as an isosceles triangle we can see the symmetry of every row as shown below:
1;
1, 2, 1;
1, 2, 0, 2, 1;
1, 2, 3, 4, 3, 2, 1;
1, 2, 3, 0, 0, 0, 3, 2, 1;
1, 2, 3, 4, 5, 7, 5, 4, 3, 2, 1;
1, 2, 3, 4, 0, 0, 0, 0, 0, 4, 3, 2, 1;
1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 2, 1;
1, 2, 3, 4, 5, 0, 0, 1, 4, 1, 0, 0, 5, 4, 3, 2, 1;
1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 9, 8, 7, 6, 5, 4, 3, 2, 1;
1, 2, 3, 4, 5, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 5, 4, 3, 2, 1;
...
For n = 15 the ziggurat diagram of 15 looks like this:
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_ | | _
_| | _| |_ | |_
_| | | | | |_
_| | | | | |_
_| | _| |_ | |_
_| | | | | |_
_| | | | | |_
_| | _| _ |_ | |_
|_ _ _ _ _ _ _ _|_ _ _|_ _ _|_|_ _ _|_ _ _|_ _ _ _ _ _ _ _|
1 2 3 4 5 6 7 8 0 0 0 1 4 7 B 7 4 1 0 0 0 8 7 6 5 4 3 2 1
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Where B = 10 + 1 = 11.
The left-hand part (or the left-hand staircase) has 8 steps.
The right-hand part (or the right-hand staircase) has 8 steps.
The central part (formed by two subparts or two staircases) has a total of 7 + 1 = 8 steps.
The number of parts equals A237271(15) = 3.
The number of subparts equals A001227(15) = 4.
The number of steps in the central column equals A067742(15) = 2.
The total number of steps equals A000203(15) = 24.
Compare the above diagram with the symmetric representation of sigma(15) with subparts as shown below:
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_ _ _|_|
_ _| | 8
| _ _|
_| |_|
|_ _| 1
| 7
_ _ _ _ _ _ _ _|
|_ _ _ _ _ _ _ _|
8
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The left-hand part has 8 square cells.
The right-hand part has 8 square cells.
The central part (formed by two subparts) has a total of 7 + 1 = 8 square cells.
The number of parts equals A237271(15) = 3.
The number of subparts equals A001227(15) = 4.
The number of square cells on the main diagonal equals A067742(15) = 2.
The total number of square cells equals A000203(15) = 24.
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