A240020 Triangle read by rows in which row n lists the parts of the symmetric representation of sigma(2n-1).

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

Author

Omar E. Pol, Mar 31 2014

Keywords

• nonn
• tabf

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

Examples

			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.
		

Crossrefs

Cf. A000203, A005408, A008438, A112610, A196020, A236104, A237048, A237270, A237271, A237591, A237593, A239053, A239660, A239931, A239933, A244050, A245092, A262626.