OFFSET
1,2
EXAMPLE
Triangle begins (first 15 rows):
1;
3;
4;
7;
6;
11, 1;
8;
15;
13;
18;
12;
23, 5;
14;
24;
23, 1;
...
For n = 12 we have that the 11th row of triangle A237593 is [6, 3, 1, 1, 1, 1, 3, 6] and the 12th row of the same triangle is [7, 2, 2, 1, 1, 2, 2, 7], so the diagram of the symmetric representation of sigma(12) = 28 is constructed as shown below in Figure 1:
. _ _
. | | | |
. | | | |
. | | | |
. | | | |
. | | | |
. _ _ _| | _ _ _| |
. 28 _| _ _| 23 _| _ _ _|
. _| | _| _| |
. | _| | _| _|
. | _ _| | |_ _|
. _ _ _ _ _ _| | _ _ _ _ _ _| | 5
. |_ _ _ _ _ _ _| |_ _ _ _ _ _ _|
.
. Figure 1. The symmetric Figure 2. After the dissection
. representation of sigma(12) of the symmetric representation
. has only one part which of sigma(12) into layers of
. contains 28 cells, so width 1 we can see two "subparts"
. A000203(12) = 28. that contain 23 and 5 cells
. respectively, so the 12th row of
. this triangle is [23, 5].
.
For n = 15 we have that the 14th row of triangle A237593 is [8, 3, 1, 2, 2, 1, 3, 8] and the 15th row of the same triangle is [8, 3, 2, 1, 1, 1, 1, 2, 3, 8], so the diagram of the symmetric representation of sigma(15) is constructed as shown below in Figure 3:
. _ _
. | | | |
. | | | |
. | | | |
. | | | |
. 8 | | 8 | |
. | | | |
. | | | |
. _ _ _|_| _ _ _|_|
. 8 _ _| | 7 _ _| |
. | _| | _ _|
. _| _| _| |_|
. |_ _| |_ _| 1
. 8 | 8 |
. _ _ _ _ _ _ _ _| _ _ _ _ _ _ _ _|
. |_ _ _ _ _ _ _ _| |_ _ _ _ _ _ _ _|
.
. Figure 3. The symmetric Figure 4. After the dissection
. representation of sigma(15) of the symmetric representation
. has three parts of size 8, of sigma(15) into layers of
. whose sum is 8 + 8 + 8 = 24, width 1 we can see four "subparts".
. so A000203(15) = 24. The first layer has three subparts
. whose sum is 8 + 7 + 8 = 23. The
. second layer has only one subpart
. of size 1, so the 15th row of this
. triangle is [23, 1].
.
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Dec 12 2016
STATUS
approved