Illustration of initial terms:
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a(n) Diagram
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0 _
1 |_|\ _
2 \ _| |\ _
3 |_ _| | |\ _
4 \ _ _|_| | |\ _
4 |_ _| _| | | |\ _
6 \ _ _| _| | | | |\ _
7 |_ _ _| _|_| | | | |\ _
8 \ _ _ _| _ _| | | | | |\ _
6 |_ _ _| | _| | | | | | |\ _
10 \ _ _ _| _| _|_| | | | | | |\ _
12 |_ _ _ _| _| _ _| | | | | | | |\ _
12 \ _ _ _ _| _| _ _| | | | | | | | |\ _
8 |_ _ _ _| | _| _ _|_| | | | | | | | |\ _
14 \ _ _ _ _| | _| | _ _| | | | | | | | | |\ _
15 |_ _ _ _ _| |_ _| | _ _| | | | | | | | | | |\ _
16 \ _ _ _ _ _| _ _|_| _ _|_| | | | | | | | | | |\
13 |_ _ _ _ _| | _| _| _ _ _| | | | | | | | | | |
18 \ _ _ _ _ _| | _| _| _ _| | | | | | | | | |
18 |_ _ _ _ _ _| | _| | _ _|_| | | | | | | |
20 \ _ _ _ _ _ _| | _| | _ _ _| | | | | | |
12 |_ _ _ _ _ _| | _ _| _| | _ _ _| | | | | |
22 \ _ _ _ _ _ _| | _ _| _|_| _ _ _|_| | | |
28 |_ _ _ _ _ _ _| | _ _| _ _| | _ _ _| | |
24 \ _ _ _ _ _ _ _| | _| | _| | _ _ _| |
14 |_ _ _ _ _ _ _| | | _| _| _| | _ _ _|
26 \ _ _ _ _ _ _ _| | |_ _| _| _| |
24 |_ _ _ _ _ _ _ _| | _ _| _| _|
28 \ _ _ _ _ _ _ _ _| | _ _| _|
24 |_ _ _ _ _ _ _ _| | | _ _|
30 \ _ _ _ _ _ _ _ _| | |
31 |_ _ _ _ _ _ _ _ _| |
32 \ _ _ _ _ _ _ _ _ _|
...
a(n) is the total area of the n-th set of symmetric regions in the diagram.
.
From Omar E. Pol, Aug 21 2015: (Start)
The above structure contains a hidden pattern, simpler, as shown below:
Level _ _
1 _| | |_
2 _| _|_ |_
3 _| | | | |_
4 _| _| | |_ |_
5 _| | _|_ | |_
6 _| _| | | | |_ |_
7 _| | | | | | |_
8 _| _| _| | |_ |_ |_
9 _| | | _|_ | | |_
10 _| _| | | | | | |_ |_
11 _| | _| | | | |_ | |_
12 _| _| | | | | | |_ |_
13 _| | | _| | |_ | | |_
14 _| _| _| | _|_ | |_ |_ |_
15 _| | | | | | | | | | |_
16 | | | | | | | | | | |
...
The symmetric pattern emerges from the front view of the stepped pyramid.
Note that starting from this diagram A000203 is obtained as follows:
In the pyramid the area of the k-th vertical region in the n-th level on the front view is equal to A237593(n,k), and the sum of all areas of the vertical regions in the n-th level on the front view is equal to 2n.
The area of the k-th horizontal region in the n-th level is equal to A237270(n,k), and the sum of all areas of the horizontal regions in the n-th level is equal to sigma(n) = A000203(n).
(End)
From Omar E. Pol, Dec 31 2016: (Star)
Illustration of the top view of the pyramid with 16 levels:
.
n A000203 A237270 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
1 1 = 1 |_| | | | | | | | | | | | | | | |
2 3 = 3 |_ _|_| | | | | | | | | | | | | |
3 4 = 2 + 2 |_ _| _|_| | | | | | | | | | | |
4 7 = 7 |_ _ _| _|_| | | | | | | | | |
5 6 = 3 + 3 |_ _ _| _| _ _|_| | | | | | | |
6 12 = 12 |_ _ _ _| _| | _ _|_| | | | | |
7 8 = 4 + 4 |_ _ _ _| |_ _|_| _ _|_| | | |
8 15 = 15 |_ _ _ _ _| _| | _ _ _|_| |
9 13 = 5 + 3 + 5 |_ _ _ _ _| | _|_| | _ _ _|
10 18 = 9 + 9 |_ _ _ _ _ _| _ _| _| |
11 12 = 6 + 6 |_ _ _ _ _ _| | _| _| _|
12 28 = 28 |_ _ _ _ _ _ _| |_ _| _|
13 14 = 7 + 7 |_ _ _ _ _ _ _| | _ _|
14 24 = 12 + 12 |_ _ _ _ _ _ _ _| |
15 24 = 8 + 8 + 8 |_ _ _ _ _ _ _ _| |
16 31 = 31 |_ _ _ _ _ _ _ _ _|
... (End)
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