A239931 Triangle read by rows in which row n lists the parts of the symmetric representation of sigma(4n-3).

1, 3, 3, 5, 3, 5, 7, 7, 9, 9, 11, 5, 5, 11, 13, 5, 13, 15, 15, 17, 7, 7, 17, 19, 19, 21, 21, 23, 32, 23, 25, 7, 25, 27, 27, 29, 11, 11, 29, 31, 31, 33, 9, 9, 33, 35, 13, 13, 35, 37, 37, 39, 18, 39, 41, 15, 9, 15, 41, 43, 11, 11, 43, 45, 45, 47, 17, 17, 47, 49, 49, 51, 51, 53, 43, 43, 53, 55, 55, 57, 57, 59, 21, 22, 21, 59, 61, 11, 61, 63, 15, 15, 63

Offset: 1

Author

Omar E. Pol, Mar 29 2014

Keywords

• nonn
• tabf

Comments

Row n is a palindromic composition of sigma(4n-3).

Row n is also the row 4n-3 of A237270.

Row n has length A237271(4n-3).

Row sums give A112610.

Also row n lists the parts of the symmetric representation of sigma in the n-th arm of the first quadrant of the spiral described in A239660, see example.

For the parts of the symmetric representation of sigma(4n-2), see A239932.

For the parts of the symmetric representation of sigma(4n-1), see A239933.

For the parts of the symmetric representation of sigma(4n), see A239934.

We can find the spiral (mentioned above) on the terraces of the pyramid described in A244050. - Omar E. Pol, Dec 06 2016

Examples

			The irregular triangle begins:
   1;
   3,  3;
   5,  3,  5;
   7,  7;
   9,  9;
  11,  5,  5, 11;
  13,  5, 13;
  15, 15;
  17,  7,  7, 17;
  19, 19;
  21, 21;
  23, 32, 23;
  25,  7, 25;
  27, 27;
  29, 11, 11, 29;
  31, 31;
  ...
Illustration of initial terms in the first quadrant of the spiral described in A239660:
.
.     _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 15
.    |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.                                  |
.                                  |
.     _ _ _ _ _ _ _ _ _ _ _ _ _ 13 |
.    |_ _ _ _ _ _ _ _ _ _ _ _ _|   |
.                              |   |_ _ _
.                              |         |
.     _ _ _ _ _ _ _ _ _ _ _ 11 |         |_
.    |_ _ _ _ _ _ _ _ _ _ _|   |_ _ _      |_
.                          |         |_ _ 5  |_
.                          |         |_  |_    |_ _
.     _ _ _ _ _ _ _ _ _ 9  |_ _ _      |_  |       |
.    |_ _ _ _ _ _ _ _ _|   |_ _  |_ 5    |_|_      |
.                      |       |_ _|_ 5      |     |_ _ _ _ _ _ 15
.                      |           | |_      |               | |
.     _ _ _ _ _ _ _ 7  |_ _        |_  |     |_ _ _ _ _ 13   | |
.    |_ _ _ _ _ _ _|       |_        | |             | |     | |
.                  |         |_      |_|_ _ _ _ 11   | |     | |
.                  |_ _        |             | |     | |     | |
.     _ _ _ _ _ 5      |_      |_ _ _ _ 9    | |     | |     | |
.    |_ _ _ _ _|         |           | |     | |     | |     | |
.              |_ _ 3    |_ _ _ 7    | |     | |     | |     | |
.              |_  |         | |     | |     | |     | |     | |
.     _ _ _ 3    |_|_ _ 5    | |     | |     | |     | |     | |
.    |_ _ _|         | |     | |     | |     | |     | |     | |
.          |_ _ 3    | |     | |     | |     | |     | |     | |
.            | |     | |     | |     | |     | |     | |     | |
.     _ 1    | |     | |     | |     | |     | |     | |     | |
.    |_|     |_|     |_|     |_|     |_|     |_|     |_|     |_|
.
For n = 7 we have that 4*7-3 = 25 and the 25th row of A237593 is [13, 5, 3, 1, 2, 1, 1, 2, 1, 3, 5, 13] and the 24th row of A237593 is [13, 4, 3, 2, 1, 1, 1, 1, 2, 3, 4, 13] therefore between both Dyck paths there are three regions (or parts) of sizes [13, 5, 13], so row 7 is [13, 5, 13].
The sum of divisors of 25 is 1 + 5 + 25 = A000203(25) = 31. On the other hand the sum of the parts of the symmetric representation of sigma(25) is 13 + 5 + 13 = 31, equaling the sum of divisors of 25.
		

Crossrefs

Cf. A000203, A112610, A196020, A236104, A235791, A237270, A237271, A237591, A237593, A239660, A239932-A239934, A244050, A245092, A262626.