For n = 1..6, consider the modular table of partitions for the first six positive integers as shown below in the fourth quadrant of the square grid (see Figure 1):
|--------------|-----------------------------------------------------|
| Modular table| Sections |
| of partitions|-----------------------------------------------------|
| for n=1..6 | 1 2 3 4 5 6 |
1--------------|-----------------------------------------------------|
. _ _ _ _ _ _ _ _ _ _ _ _
. |_| | | | | | |_| _| | | | | | | | | |
. |_ _| | | | | |_ _| _ _| | | | | | | |
. |_ _ _| | | | |_ _ _| _ _ _| | | | | |
. |_ _| | | | |_ _| | | | | |
. |_ _ _ _| | | |_ _ _ _| _ _ _ _| | | |
. |_ _ _| | | |_ _ _| | | |
. |_ _ _ _ _| | |_ _ _ _ _| _ _ _ _ _| |
. |_ _| | | |_ _| | |
. |_ _ _ _| | |_ _ _ _| |
. |_ _ _| | |_ _ _| |
. |_ _ _ _ _ _| |_ _ _ _ _ _|
.
. Figure 1. Figure 2.
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The table contains 11 regions, see Figure 1.
The regions are distributed in 6 sections. The Figure 2 shows the sections separately.
Then consider the following table which contains the diagram of every region separately:
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| | | | | | |
| Section | Region | Parts | Region | Peri- | a(n) |
| | |(A220482)| diagram | meter | |
---------------------------------------------------------------------
| | | | _ | | |
| 1 | 1 | 1 | |_| | 4 | 4 |
---------------------------------------------------------------------
| | | | _ | | |
| | | 1 | _| | | | |
| 2 | 2 | 2 | |_ _| | 8 | 8 |
---------------------------------------------------------------------
| | | | _ | | |
| | | 1 | | | | | |
| | | 1 | _ _| | | | |
| 3 | 3 | 3 | |_ _ _| | 12 | 12 |
---------------------------------------------------------------------
| | | | _ _ | | |
| | 4 | 2 | |_ _| | 6 | |
| |---------|---------|----------------------------| |
| | | | _ | | |
| | | 1 | | | | | |
| | | 1 | | | | | |
| | | 1 | _| | | | |
| | | 2 | _ _| | | | |
| 4 | 5 | 4 | |_ _ _ _| | 18 | 24 |
---------------------------------------------------------------------
| | | | _ _ _ | | |
| | 6 | 3 | |_ _ _| | 8 | |
| |---------|---------|--------------------|-------| |
| | | | _ | | |
| | | 1 | | | | | |
| | | 1 | | | | | |
| | | 1 | | | | | |
| | | 1 | | | | | |
| | | 1 | _| | | | |
| | | 2 | _ _ _| | | | |
| 5 | 7 | 5 | |_ _ _ _ _| | 24 | 32 |
---------------------------------------------------------------------
| | | | _ _ | | |
| | 8 | 2 | |_ _| | 6 | |
| |---------|---------|--------------------|-------| |
| | | | _ _ | | |
| | | 2 | _ _| | | | |
| | 9 | 4 | |_ _ _ _| | 12 | |
1 |---------|---------|--------------------|-------| |
| | | | _ _ _ | | |
| | 10 | 3 | |_ _ _| | 8 | |
| |---------|---------|--------------------|-------| |
| | | | _ | | |
| | | 1 | | | | | |
| | | 1 | | | | | |
| | | 1 | | | | | |
| | | 1 | | | | | |
| | | 1 | | | | | |
| | | 1 | | | | | |
| | | 1 | _| | | | |
| | | 2 | | | | | |
| | | 2 | _| | | | |
| | | 3 | _ _ _| | | | |
| 6 | 11 | 6 | |_ _ _ _ _ _| | 34 | 60 |
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For n = 1..3, there is only one region in every section. The perimeters of the regions are 4, 8 and 12 respectively, so a(1) = 4, a(2) = 8, and a(3) = 12.
For n = 4, the 4th section contains two regions with perimeters 6 and 18 respectively. The sum of the perimeters is 6 + 18 = 24, so a(4) = 24.
For n = 5, the 5th section contains two regions with perimeters 8 and 24 respectively. The sum of the perimeters is 8 + 24 = 32, so a(5) = 32.
For n = 6, the 6th section contains four regions with perimeters 6, 12, 8 and 34 respectively. The sum of the perimeters is 6 + 12 + 8 + 34 = 60, so a(6) = 60.
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