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A240020
Triangle read by rows in which row n lists the parts of the symmetric representation of sigma(2n-1).
4
1, 2, 2, 3, 3, 4, 4, 5, 3, 5, 6, 6, 7, 7, 8, 8, 8, 9, 9, 10, 10, 11, 5, 5, 11, 12, 12, 13, 5, 13, 14, 6, 6, 14, 15, 15, 16, 16, 17, 7, 7, 17, 18, 12, 18, 19, 19, 20, 8, 8, 20, 21, 21, 22, 22, 23, 32, 23, 24, 24, 25, 7, 25, 26, 10, 10, 26, 27, 27, 28, 8, 8, 28, 29, 11, 11, 29, 30, 30, 31, 31, 32, 12, 26, 12, 32, 33, 9, 9, 33, 34, 34
OFFSET
1,2
COMMENTS
Row n lists the parts of the symmetric representation of A008438(n-1).
Also these are the parts from the odd-indexed rows of A237270.
Also these are the parts in the quadrants 1 and 3 of the spiral described in A239660, see example.
Row sums give A008438.
The length of row n is A237271(2n-1).
Both column 1 and the right border are equal to n.
Note that also the sequence can be represented in a quadrant.
We can find the spiral (mentioned above) on the terraces of the stepped pyramid described in A244050. - Omar E. Pol, Dec 07 2016
EXAMPLE
1;
2, 2;
3, 3;
4, 4;
5, 3, 5;
6, 6;
7, 7;
8, 8, 8;
9, 9;
10, 10;
11, 5, 5, 11;
12, 12;
13, 5, 13;
14, 6, 6, 14;
15, 15;
16, 16;
17, 7, 7, 17;
18, 12, 18;
19, 19;
20, 8, 8, 20;
21, 21;
22, 22;
23, 32, 23;
24, 24;
25, 7, 25;
...
Illustration of initial terms (rows 1..8):
.
. _ _ _ _ _ _ _ 7
. |_ _ _ _ _ _ _|
. |
. |_ _
. _ _ _ _ _ 5 |_
. |_ _ _ _ _| |
. |_ _ 3 |_ _ _ 7
. |_ | | |
. _ _ _ 3 |_|_ _ 5 | |
. |_ _ _| | | | |
. |_ _ 3 | | | |
. | | | | | |
. _ 1 | | | | | |
. _ _ _ _ |_| |_| |_| |_|
. | | | | | | | |
. | | | | | | |_|_ _
. | | | | | | 2 |_ _|
. | | | | |_|_ 2
. | | | | 4 |_
. | | |_|_ _ |_ _ _ _
. | | 6 |_ |_ _ _ _|
. |_|_ _ _ |_ 4
. 8 | |_ _ |
. |_ | |_ _ _ _ _ _
. |_ |_ |_ _ _ _ _ _|
. 8 |_ _| 6
. |
. |_ _ _ _ _ _ _ _
. |_ _ _ _ _ _ _ _|
. 8
.
The figure shows the quadrants 1 and 3 of the spiral described in A239660.
For n = 5 we have that 2*5 - 1 = 9 and the 9th row of A237593 is [5, 2, 2, 2, 2, 5] and the 8th row of A237593 is [5, 2, 1, 1, 2, 5] therefore between both symmetric Dyck paths there are three regions (or parts) of sizes [5, 3, 5], so row 5 is [5, 3, 5], see the third arm of the spiral in the first quadrant.
The sum of divisors of 9 is 1 + 3 + 9 = A000203(9) = 13. On the other hand the sum of the parts of the symmetric representation of sigma(9) is 5 + 3 + 5 = 13, equaling the sum of divisors of 9.
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Mar 31 2014
STATUS
approved