A237590 -id:A237590 - cp's OEIS frontend

A340846 a(n) is the number of edges in the diagram of the symmetric representation of sigma(n).

4, 6, 8, 10, 10, 12, 12, 14, 16, 16, 14, 18, 14, 18, 22, 22, 16, 22, 16, 22, 26, 22, 18, 26, 24, 22, 28, 28, 20, 30, 20, 30, 30, 24, 28, 32, 22, 26, 32, 34, 22, 34, 22, 34, 38, 28, 24, 38, 32, 40, 34, 36, 24, 38, 38, 42, 36, 30, 26, 42, 26, 30, 46, 42, 40, 44, 28

Offset: 1

Omar E. Pol, Jan 24 2021

nonn

Since the diagram is symmetric so all terms are even numbers.

For another version with subparts see A340848 from which first differs at a(6).

			Illustration of initial terms:
.                                                          _ _ _ _
.                                            _ _ _        |_ _ _  |_
.                                _ _ _      |_ _ _|             |   |_
.                      _ _      |_ _  |_          |_ _          |_ _  |
.              _ _    |_ _|_        |_  |           | |             | |
.        _    |_  |       | |         | |           | |             | |
.       |_|     |_|       |_|         |_|           |_|             |_|
.
n:       1      2        3          4           5               6
a(n):    4      6        8         10          10              12
.
For n = 6 the diagram has 12 edges so a(6) = 12.
On the other hand the diagram has 12 vertices and only one part or region, so applying Euler's formula we have that a(6) = 12 + 1 - 1 = 12.
.                                                  _ _ _ _ _
.                            _ _ _ _ _            |_ _ _ _ _|
.        _ _ _ _            |_ _ _ _  |                     |_ _
.       |_ _ _ _|                   | |_                    |_  |
.               |_                  |_  |_ _                  |_|_ _
.                 |_ _                |_ _  |                     | |
.                   | |                   | |                     | |
.                   | |                   | |                     | |
.                   | |                   | |                     | |
.                   |_|                   |_|                     |_|
.
n:              7                    8                      9
a(n):          12                   14                     16
.
For n = 9 the diagram has 16 edges so a(9) = 16.
On the other hand the diagram has 14 vertices and three parts or regions, so applying Euler's formula we have that a(9) = 14 + 3 - 1 = 16.
Another way for the illustration of initial terms is as follows:
--------------------------------------------------------------------------
.  n  a(n)                             Diagram
--------------------------------------------------------------------------
            _
   1   4   |_|  _
              _| |  _
   2   6     |_ _| | |  _
                _ _|_| | |  _
   3   8       |_ _|  _| | | |  _
                  _ _|  _| | | | |  _
   4  10         |_ _ _|  _|_| | | | |  _
                    _ _ _|  _ _| | | | | |  _
   5  10           |_ _ _| |    _| | | | | | |  _
                      _ _ _|  _|  _|_| | | | | | |  _
   6  12             |_ _ _ _|  _|  _ _| | | | | | | |  _
                        _ _ _ _|  _|  _ _| | | | | | | | |  _
   7  12               |_ _ _ _| |  _|  _ _|_| | | | | | | | |  _
                          _ _ _ _| |  _| |  _ _| | | | | | | | | |  _
   8  14                 |_ _ _ _ _| |_ _| |  _ _| | | | | | | | | | |  _
                            _ _ _ _ _|  _ _|_|  _ _|_| | | | | | | | | | |
   9  16                   |_ _ _ _ _| |  _|  _|  _ _ _| | | | | | | | | |
                              _ _ _ _ _| |  _|  _|    _ _| | | | | | | | |
  10  16                     |_ _ _ _ _ _| |  _|     |  _ _|_| | | | | | |
                                _ _ _ _ _ _| |      _| |  _ _ _| | | | | |
  11  14                       |_ _ _ _ _ _| |  _ _|  _| |  _ _ _| | | | |
                                  _ _ _ _ _ _| |  _ _|  _|_|  _ _ _|_| | |
  12  18                         |_ _ _ _ _ _ _| |  _ _|  _ _| |  _ _ _| |
                                    _ _ _ _ _ _ _| |  _| |    _| |  _ _ _|
  13  14                           |_ _ _ _ _ _ _| | |  _|  _|  _| |
                                      _ _ _ _ _ _ _| | |_ _|  _|  _|
  14  18                             |_ _ _ _ _ _ _ _| |  _ _|  _|
                                        _ _ _ _ _ _ _ _| |  _ _|
  15  22                               |_ _ _ _ _ _ _ _| | |
                                          _ _ _ _ _ _ _ _| |
  16  22                                 |_ _ _ _ _ _ _ _ _|
...

Cf. A237271 (number of parts or regions).
Cf. A340833 (number of vertices).
Cf. A340848 (number of edges in the diagram with subparts).
Cf. A317109 (total number of edges in the unified diagram).
Cf. A239931-A239934 (illustration of first 32 diagrams).
Cf. A000203, A005843, A196020, A236104, A235791, A237048, A237270, A237590, A237591, A237593, A239660, A245092, A262626, A340847.

a(n) = A340833(n) + A237271(n) - 1 (Euler's formula).

More terms from Omar E. Pol, Oct 28 2021

A294847 a(n) is the total number of line segments in the diagram described in A236104 after n-th stage, with a(0) = 2.

Original entry on oeis.org

2, 4, 6, 8, 12, 14, 20, 22, 28, 32, 38, 40, 48, 50, 56, 64, 74, 76, 86, 88, 98, 106, 112, 114, 126, 132, 138, 146, 160, 162

Offset: 0

Omar E. Pol, Nov 09 2017

Note that the diagram is also the top view of the stepped pyramid with n levels described in A245092.

For the construction of the diagram using Dyck paths and for more information about the pyramid see A237593.

			Illustration of initial terms (n = 1..6):
.                                                      _ _ _ _
.                                        _ _ _        |_ _ _  |_
.                            _ _ _      |_ _ _|       |_ _ _|   |_
.                  _ _      |_ _  |_    |_ _  |_ _    |_ _  |_ _  |
.          _ _    |_ _|_    |_ _|_  |   |_ _|_  | |   |_ _|_  | | |
.    _    |_  |   |_  | |   |_  | | |   |_  | | | |   |_  | | | | |
.   |_|   |_|_|   |_|_|_|   |_|_|_|_|   |_|_|_|_|_|   |_|_|_|_|_|_|
.
.    4      6        8          12           14             20

Cf. A000203, A024916, A196020, A235791, A236104, A237048, A237270, A237271, A237590, A237591, A237593, A245092, A262626, A294723.

A299778 Irregular triangle read by rows: T(n,k) is the part that is adjacent to the k-th peak of the largest Dyck path of the symmetric representation of sigma(n), or T(n,k) = 0 if the mentioned part is already associated to a previous peak or if there is no part adjacent to the k-th peak, with n >= 1, k >= 1.

Original entry on oeis.org

1, 3, 2, 2, 7, 0, 3, 3, 12, 0, 0, 4, 0, 4, 15, 0, 0, 5, 3, 5, 9, 0, 9, 0, 6, 0, 0, 6, 28, 0, 0, 0, 7, 0, 0, 7, 12, 0, 12, 0, 8, 8, 0, 0, 8, 31, 0, 0, 0, 0, 9, 0, 0, 0, 9, 39, 0, 0, 0, 0, 10, 0, 0, 0, 10, 42, 0, 0, 0, 0, 11, 5, 0, 5, 0, 11, 18, 0, 0, 0, 18, 0, 12, 0, 0, 0, 0, 12, 60, 0, 0, 0, 0, 0, 13, 0, 5, 0, 0, 13

Offset: 1

Omar E. Pol, Apr 03 2018

For the definition of "part" of the symmetric representation of sigma see A237270.

For more information about the mentioned Dyck paths see A237593.

			Triangle begins (rows 1..28):
   1;
   3;
   2,  2;
   7,  0;
   3,  3;
  12,  0,  0;
   4,  0,  4;
  15,  0,  0;
   5,  3,  5;
   9,  0,  9,  0;
   6,  0,  0,  6;
  28,  0,  0,  0;
   7,  0,  0,  7;
  12,  0, 12,  0;
   8,  8,  0,  0,  8;
  31,  0,  0,  0,  0;
   9,  0,  0,  0,  9;
  39,  0,  0,  0,  0;
  10,  0,  0,  0, 10;
  42,  0,  0,  0,  0;
  11,  5,  0,  5,  0, 11;
  18,  0,  0,  0, 18,  0;
  12,  0,  0,  0,  0, 12;
  60,  0,  0,  0,  0,  0;
  13,  0,  5,  0,  0, 13;
  21,  0,  0,  0  21,  0;
  14,  6,  0,  6,  0, 14;
  56,  0,  0,  0,  0,  0,  0;
  ...
Illustration of first 50 terms (rows 1..16 of triangle) in an irregular spiral which can be find in the top view of the pyramid described in A244050:
.
.               12 _ _ _ _ _ _ _ _
.                 |  _ _ _ _ _ _ _|_ _ _ _ _ _ _ 7
.                 | |             |_ _ _ _ _ _ _|
.              0 _| |                           |
.               |_ _|9 _ _ _ _ _ _              |_ _ 0
.         12 _ _|     |  _ _ _ _ _|_ _ _ _ _ 5      |_ 0
.    0 _ _ _| |    0 _| |         |_ _ _ _ _|         |
.     |  _ _ _|  9 _|_ _|                   |_ _ 3    |_ _ _ 7
.     | |    0 _ _| |   12 _ _ _ _          |_  |         | |
.     | |     |  _ _|  0 _|  _ _ _|_ _ _ 3    |_|_ _ 5    | |
.     | |     | |    0 _|   |     |_ _ _|         | |     | |
.     | |     | |     |  _ _|           |_ _ 3    | |     | |
.     | |     | |     | |    3 _ _        | |     | |     | |
.     | |     | |     | |     |  _|_ 1    | |     | |     | |
.    _|_|    _|_|    _|_|    _|_| |_|    _|_|    _|_|    _|_|    _
.   | |     | |     | |     | |         | |     | |     | |     | |
.   | |     | |     | |     |_|_ _     _| |     | |     | |     | |
.   | |     | |     | |    2  |_ _|_ _|  _|     | |     | |     | |
.   | |     | |     |_|_     2    |_ _ _|  0 _ _| |     | |     | |
.   | |     | |    4    |_               7 _|  _ _|0    | |     | |
.   | |     |_|_ _     0  |_ _ _ _        |  _|    _ _ _| |     | |
.   | |    6      |_      |_ _ _ _|_ _ _ _| |  0 _|    _ _|0    | |
.   |_|_ _ _     0  |_   4        |_ _ _ _ _|  _|     |    _ _ _| |
.  8      | |_ _   0  |                     15|      _|   |  _ _ _|
.         |_    |     |_ _ _ _ _ _            |  _ _|  0 _| |      0
.        8  |_  |_    |_ _ _ _ _ _|_ _ _ _ _ _| |    0 _|  _|
.          0  |_ _|  6            |_ _ _ _ _ _ _|  _ _|  _|  0
.            0    |                             28|  _ _|  0
.                 |_ _ _ _ _ _ _ _                | |    0
.                 |_ _ _ _ _ _ _ _|_ _ _ _ _ _ _ _| |
.                8                |_ _ _ _ _ _ _ _ _|
.                                                    31
.
The diagram contains A237590(16) = 27 parts.
For the construction of the spiral see A239660.

Index entries for sequences related to sigma(n)

Row sums give A000203.
Row n has length A003056(n).
Column k starts in row A000217(k).
Nonzero terms give A237270.
The number of nonzero terms in row n is A237271(n).
Column 1 is A241838.
The triangle with n rows contain A237590(n) nonzero terms.
Cf. A296508 (analog for subparts).
Cf. A024916, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A239657, A239660, A239931-A239934, A240542, A244050, A245092, A250068, A250070, A261699, A262626, A279387, A279388, A279391, A280850, A280851.

A317292 a(n) is the total number of edges after n-th stage in the diagram of the symmetries of sigma in which the parts of width > 1 are dissected into subparts of width 1, with a(0) = 0.

Original entry on oeis.org

0, 4, 8, 14, 20, 26, 36, 42, 50, 60, 70, 76, 92, 98, 108, 124, 136, 142, 160, 166, 182, 198, 208, 214, 238, 250, 260, 276, 294, 300

Offset: 0

Omar E. Pol, Jul 27 2018

All terms are even numbers.

Note that in the diagram the number of regions or subparts equals A060831, the partial sums of A001227, n >= 1.

			Illustration of initial terms (n = 1..9):
.                                                       _ _ _ _
.                                         _ _ _        |_ _ _  |_
.                             _ _ _      |_ _ _|       |_ _ _| |_|_
.                   _ _      |_ _  |_    |_ _  |_ _    |_ _  |_ _  |
.           _ _    |_ _|_    |_ _|_  |   |_ _|_  | |   |_ _|_  | | |
.     _    |_  |   |_  | |   |_  | | |   |_  | | | |   |_  | | | | |
.    |_|   |_|_|   |_|_|_|   |_|_|_|_|   |_|_|_|_|_|   |_|_|_|_|_|_|
.
.     4      8        14         20           26             36
.
.                                               _ _ _ _ _
.                         _ _ _ _ _            |_ _ _ _ _|
.     _ _ _ _            |_ _ _ _  |           |_ _ _ _  |_ _
.    |_ _ _ _|           |_ _ _ _| |_          |_ _ _ _| |_  |
.    |_ _ _  |_          |_ _ _  |_  |_ _      |_ _ _  |_  |_|_ _
.    |_ _ _| |_|_ _      |_ _ _| |_|_ _  |     |_ _ _| |_|_ _  | |
.    |_ _  |_ _  | |     |_ _  |_ _  | | |     |_ _  |_ _  | | | |
.    |_ _|_  | | | |     |_ _|_  | | | | |     |_ _|_  | | | | | |
.    |_  | | | | | |     |_  | | | | | | |     |_  | | | | | | | |
.    |_|_|_|_|_|_|_|     |_|_|_|_|_|_|_|_|     |_|_|_|_|_|_|_|_|_|
.
.           42                  50                     60
.
.
Illustration of the two-dimensional diagram after 29 stages (contains 300 edges, 239 vertices and 62 regions or subparts):
._ _ _ _ _ _ _ _ _ _ _ _ _ _ _
|_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
|_ _ _ _ _ _ _ _ _ _ _ _ _ _  |
|_ _ _ _ _ _ _ _ _ _ _ _ _ _| |
|_ _ _ _ _ _ _ _ _ _ _ _ _  | |
|_ _ _ _ _ _ _ _ _ _ _ _ _| | |
|_ _ _ _ _ _ _ _ _ _ _ _  | | |_ _ _
|_ _ _ _ _ _ _ _ _ _ _ _| | |_ _ _  |
|_ _ _ _ _ _ _ _ _ _ _  | | |_ _  | |_
|_ _ _ _ _ _ _ _ _ _ _| | |_ _ _| |_  |_
|_ _ _ _ _ _ _ _ _ _  | | |_ _  |_ _| |_|_
|_ _ _ _ _ _ _ _ _ _| | |_ _  | |_  |_ _  |_ _
|_ _ _ _ _ _ _ _ _  | |_ _ _| |_  |_  | |_ _  |
|_ _ _ _ _ _ _ _ _| | |_ _  |_  |_  |_|_ _  | |
|_ _ _ _ _ _ _ _  | |_ _  |_ _|_  |_ _  | | | |_ _ _ _ _ _
|_ _ _ _ _ _ _ _| | |_ _| |_  | |_ _  | | |_|_ _ _ _ _  | |
|_ _ _ _ _ _ _  | |_ _  |_  |_|_  | | |_|_ _ _ _ _  | | | |
|_ _ _ _ _ _ _| |_ _  |_  |_ _  | | |_ _ _ _ _  | | | | | |
|_ _ _ _ _ _  | |_  |_  |_  | | |_|_ _ _ _  | | | | | | | |
|_ _ _ _ _ _| |_ _| |_|_  | |_|_ _ _ _  | | | | | | | | | |
|_ _ _ _ _  | |_  |_ _  | |_ _ _ _  | | | | | | | | | | | |
|_ _ _ _ _| |_  |_  | |_|_ _ _  | | | | | | | | | | | | | |
|_ _ _ _  |_ _|_  |_|_ _ _  | | | | | | | | | | | | | | | |
|_ _ _ _| |_  | |_ _ _  | | | | | | | | | | | | | | | | | |
|_ _ _  |_  |_|_ _  | | | | | | | | | | | | | | | | | | | |
|_ _ _| |_|_ _  | | | | | | | | | | | | | | | | | | | | | |
|_ _  |_ _  | | | | | | | | | | | | | | | | | | | | | | | |
|_ _|_  | | | | | | | | | | | | | | | | | | | | | | | | | |
|_  | | | | | | | | | | | | | | | | | | | | | | | | | | | |
|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
.

For the definition of "subparts" see A279387.
For the triangle of sums of subparts see A279388.
Cf. A317293 (number of vertices).
Cf. A060831 (number of regions or subparts).
Compare with A317109 (analog for the diagram that contains only parts).
First differs from A317109 at a(6).
Cf. A000203, A001227, A196020, A235791, A237048, A237590, A237591, A237270, A237271, A237593, A245092, A244050, A262626, A280850, A280851, A280940, A285901, A294723, A296508.

a(n) = A317293(n) + A060831(n) - 1 (Euler's formula).

A317293 a(n) is the total number of vertices after n-th stage in the diagram of the symmetries of sigma in which the parts of width > 1 are dissected into subparts of width 1, with a(0) = 1.

Original entry on oeis.org

1, 4, 7, 11, 16, 20, 28, 32, 39, 46, 54, 58, 72, 76, 84, 96, 107, 111, 126, 130, 144, 156, 164, 168, 190, 199, 207, 219, 235, 239

Offset: 0

Omar E. Pol, Jul 27 2018

Note that in the diagram the number of regions or subparts equals A060831, the partial sums of A001227, n >= 1.

			Illustration of initial terms (n = 0..9):
.                                                           _ _ _ _
.                                             _ _ _        |_ _ _  |_
.                                 _ _ _      |_ _ _|       |_ _ _| |_|_
.                       _ _      |_ _  |_    |_ _  |_ _    |_ _  |_ _  |
.               _ _    |_ _|_    |_ _|_  |   |_ _|_  | |   |_ _|_  | | |
.         _    |_  |   |_  | |   |_  | | |   |_  | | | |   |_  | | | | |
.    .   |_|   |_|_|   |_|_|_|   |_|_|_|_|   |_|_|_|_|_|   |_|_|_|_|_|_|
.
.    1    4      7        11         16           20             28
.
.                                               _ _ _ _ _
.                         _ _ _ _ _            |_ _ _ _ _|
.     _ _ _ _            |_ _ _ _  |           |_ _ _ _  |_ _
.    |_ _ _ _|           |_ _ _ _| |_          |_ _ _ _| |_  |
.    |_ _ _  |_          |_ _ _  |_  |_ _      |_ _ _  |_  |_|_ _
.    |_ _ _| |_|_ _      |_ _ _| |_|_ _  |     |_ _ _| |_|_ _  | |
.    |_ _  |_ _  | |     |_ _  |_ _  | | |     |_ _  |_ _  | | | |
.    |_ _|_  | | | |     |_ _|_  | | | | |     |_ _|_  | | | | | |
.    |_  | | | | | |     |_  | | | | | | |     |_  | | | | | | | |
.    |_|_|_|_|_|_|_|     |_|_|_|_|_|_|_|_|     |_|_|_|_|_|_|_|_|_|
.
.           32                  39                     46
.
.
Illustration of the two-dimensional diagram after 29 stages (contains 239 vertices, 300 edges and 62 regions or subparts):
._ _ _ _ _ _ _ _ _ _ _ _ _ _ _
|_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
|_ _ _ _ _ _ _ _ _ _ _ _ _ _  |
|_ _ _ _ _ _ _ _ _ _ _ _ _  | |
|_ _ _ _ _ _ _ _ _ _ _ _ _| | |
|_ _ _ _ _ _ _ _ _ _ _ _  | | |_ _ _
|_ _ _ _ _ _ _ _ _ _ _ _| | |_ _ _  |
|_ _ _ _ _ _ _ _ _ _ _  | | |_ _  | |_
|_ _ _ _ _ _ _ _ _ _ _| | |_ _ _| |_  |_
|_ _ _ _ _ _ _ _ _ _  | | |_ _  |_ _| |_|_
|_ _ _ _ _ _ _ _ _ _| | |_ _  | |_  |_ _  |_ _
|_ _ _ _ _ _ _ _ _  | |_ _ _| |_  |_  | |_ _  |
|_ _ _ _ _ _ _ _ _| | |_ _  |_  |_  |_|_ _  | |
|_ _ _ _ _ _ _ _  | |_ _  |_ _|_  |_ _  | | | |_ _ _ _ _ _
|_ _ _ _ _ _ _ _| | |_ _| |_  | |_ _  | | |_|_ _ _ _ _  | |
|_ _ _ _ _ _ _  | |_ _  |_  |_|_  | | |_|_ _ _ _ _  | | | |
|_ _ _ _ _ _ _| |_ _  |_  |_ _  | | |_ _ _ _ _  | | | | | |
|_ _ _ _ _ _  | |_  |_  |_  | | |_|_ _ _ _  | | | | | | | |
|_ _ _ _ _ _| |_ _| |_|_  | |_|_ _ _ _  | | | | | | | | | |
|_ _ _ _ _  | |_  |_ _  | |_ _ _ _  | | | | | | | | | | | |
|_ _ _ _ _| |_  |_  | |_|_ _ _  | | | | | | | | | | | | | |
|_ _ _ _  |_ _|_  |_|_ _ _  | | | | | | | | | | | | | | | |
|_ _ _ _| |_  | |_ _ _  | | | | | | | | | | | | | | | | | |
|_ _ _  |_  |_|_ _  | | | | | | | | | | | | | | | | | | | |
|_ _ _| |_|_ _  | | | | | | | | | | | | | | | | | | | | | |
|_ _  |_ _  | | | | | | | | | | | | | | | | | | | | | | | |
|_ _|_  | | | | | | | | | | | | | | | | | | | | | | | | | |
|_  | | | | | | | | | | | | | | | | | | | | | | | | | | | |
|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
.

For the definition of "subparts" see A279387.
For the triangle of sums of subparts see A279388.
Cf. A317292 (number of edges).
Cf. A060831 (number of regions or subparts).
Compare with A294723 (analog for the diagram that contains only parts).
First differs from A294723 at a(6).
Cf. A000203, A196020, A235791, A237048, A237590, A237591, A237270, A237271, A237593, A245092, A244050, A262626, A280850, A280851, A280940, A285901, A296508, A317109.

a(n) = A317292(n) - A060831(n) + 1 (Euler's formula).

A293750 a(n) is the total number of line segments that belong to the Dyck paths in the diagram of the symmetries of sigma described in A236104 and A237593 after n-th stage, with a(0) = 0.

Original entry on oeis.org

0, 2, 4, 6, 10, 12, 18, 20, 26, 30, 36, 38, 46, 48, 54, 62, 72, 74, 84, 86, 96, 104, 110, 112, 124, 130, 136, 144, 158, 160

Offset: 0

Omar E. Pol, Nov 09 2017

			Illustration of initial terms (n = 1..6):
.                                                      _ _ _ _
.                                        _ _ _         _ _ _  |_
.                            _ _ _       _ _ _|        _ _ _|   |_
.                  _ _       _ _  |_     _ _  |_ _     _ _  |_ _  |
.          _ _     _ _|_     _ _|_  |    _ _|_  | |    _ _|_  | | |
.    _     _  |    _  | |    _  | | |    _  | | | |    _  | | | | |
.     |     | |     | | |     | | | |     | | | | |     | | | | | |
.
.    2      4        6          10           12             18
.

Cf. A000203, A024916, A196020, A235791, A236104, A237048, A237270, A237271, A237590, A237591, A237593, A245092, A262626, A294723, A294847.

a(n) = A294847(n) - 2.

A339583 Leading term in row 2*n of A237270.

Original entry on oeis.org

3, 7, 12, 15, 9, 28, 12, 31, 39, 42, 18, 60, 21, 56, 72, 63, 27, 91, 30, 90, 96, 42, 36, 124, 39, 49, 120, 120, 45, 168, 48, 127, 144, 63, 54, 195, 57, 70, 84, 186, 63, 224, 66, 180, 234, 84, 72, 252, 75, 217, 108, 210, 81, 280, 84, 248, 120, 105, 90, 360, 93, 112, 312, 255, 99, 336

Offset: 1

N. J. A. Sloane, Dec 25 2020

nonn

The leading term in row 2*n-1 is n.
The first column in A237270, [1, 3, 2, 7, 3, 12, 4, 15, 5, 9, 6, 28, 7, 12, 8, 31, 9, 39, 10, 42, 11, ...], without the initial 1, is A237270(A237590(n)+1). By bisecting this we get the present sequence.
Bisection of A241838. - Omar E. Pol, Feb 23 2021

			Row 10 of A237270 is [9, 9], so a(5) = 9 (the first of the two 9's, officially).

N. J. A. Sloane, Table of n, a(n) for n = 1..2495

Cf. A237270, A237271, A237590, A237593, A241838.

A347528 Total number of layers of width 1 of all symmetric representations of sigma() with subparts of all positive integers <= n.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 19, 20, 22, 23, 25, 26, 27, 28, 30, 31, 32, 33, 35, 36, 38, 39, 40, 41, 42, 44, 46, 47, 48, 49, 51, 52, 54, 55, 56, 58, 59, 60, 62, 63, 64, 65, 66, 67, 69, 70, 72, 73, 74, 75, 78, 79, 80, 82, 83, 84, 86, 87, 88

Offset: 1

Omar E. Pol, Sep 05 2021

nonn

			For the first five positive integers every symmetric representation of sigma() with subparts has only one layer of width 1, so a(5) = 1 + 1 + 1 + 1 + 1 = 5.
For n = 6 the symmetric representation of sigma(6) with subparts has two layers of width 1 as shown below:
                     _ _ _ _
                    |_ _ _  |_
                          | |_|_
                          |_ _  |
                              | |
                              | |
                              |_|
So a(6) = 5 + 2 = 7.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Partial sums of A250068.

Cf. A196020, A235791, A236104, A237270, A237271, A237591, A237590, A237593, A279387, A279391, A279667, A280850, A280851, A296508.

Accumulate@ Map[Max@ Accumulate[#] &, Table[If[OddQ[k], Boole@ Divisible[n, k], -Boole@ Divisible[n - k/2, k]], {n, 68}, {k, Floor[(Sqrt[8 n + 1] - 1)/2]}]] (* Michael De Vlieger, Oct 27 2021 *)

A302248 Irregular triangle read by rows in which the odd-indexed terms of the n-th row together with the even-indexed terms of the same row but listed in reverse order give the n-th row of triangle A299778.

Original entry on oeis.org

1, 3, 2, 2, 7, 0, 3, 3, 12, 0, 0, 4, 4, 0, 15, 0, 0, 5, 5, 3, 9, 0, 0, 9, 6, 6, 0, 0, 28, 0, 0, 0, 7, 7, 0, 0, 12, 0, 0, 12, 8, 8, 8, 0, 0, 31, 0, 0, 0, 0, 9, 9, 0, 0, 0, 39, 0, 0, 0, 0, 10, 10, 0, 0, 0, 42, 0, 0, 0, 0, 11, 11, 5, 0, 0, 5, 18, 0, 0, 18, 0, 0, 12, 12, 0, 0, 0, 0, 60, 0, 0, 0, 0, 0, 13, 13, 0, 0, 5, 0

Offset: 1

Omar E. Pol, Apr 10 2018

			Triangle begins (rows 1..28):
   1;
   3;
   2,  2;
   7,  0;
   3,  3;
  12,  0,  0;
   4,  4,  0;
  15,  0,  0;
   5,  5,  3;
   9,  0,  0,  9;
   6,  6,  0,  0;
  28,  0,  0,  0;
   7,  7,  0,  0;
  12,  0,  0, 12;
   8,  8,  8,  0,  0;
  31,  0,  0,  0,  0;
   9,  9,  0,  0,  0;
  39,  0,  0,  0,  0;
  10, 10,  0,  0,  0;
  42,  0,  0,  0,  0;
  11, 11,  5,  0,  0,  5;
  18,  0,  0, 18,  0,  0;
  12, 12,  0,  0,  0,  0;
  60,  0,  0,  0,  0,  0;
  13, 13,  0,  0,  5,  0;
  21,  0,  0, 21,  0,  0;
  14, 14,  6,  0,  0,  6;
  56,  0,  0,  0,  0,  0,  0;
...
For n = 21 the 21st row of A299778 is [11, 5, 0, 5, 0, 11], so the 21st row of this triangle is [11, 11, 5, 0, 0, 5].

Index entries for sequences related to sigma(n)

Row sums give A000203.
Row n has length A003056(n).
Column k starts in row A000217(k).
The number of nonzero terms in row n is A237271(n).
Column 1 is A241838.
The triangle with n rows contain A237590(n) nonzero terms.
Cf. A280850 (analog for subparts).
Cf. A196020, A235791, A236104, A237048, A237270, A237591, A237593, A239660, A240542, A244050, A245092, A262626, A279387, A279388, A280851, A299778.

cp's OEIS Frontend

A340846 a(n) is the number of edges in the diagram of the symmetric representation of sigma(n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A294847 a(n) is the total number of line segments in the diagram described in A236104 after n-th stage, with a(0) = 2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A317292 a(n) is the total number of edges after n-th stage in the diagram of the symmetries of sigma in which the parts of width > 1 are dissected into subparts of width 1, with a(0) = 0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A317293 a(n) is the total number of vertices after n-th stage in the diagram of the symmetries of sigma in which the parts of width > 1 are dissected into subparts of width 1, with a(0) = 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A293750 a(n) is the total number of line segments that belong to the Dyck paths in the diagram of the symmetries of sigma described in A236104 and A237593 after n-th stage, with a(0) = 0.

Views

Author

Keywords

Examples

Crossrefs

Formula

A339583 Leading term in row 2*n of A237270.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A347528 Total number of layers of width 1 of all symmetric representations of sigma() with subparts of all positive integers <= n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A302248 Irregular triangle read by rows in which the odd-indexed terms of the n-th row together with the even-indexed terms of the same row but listed in reverse order give the n-th row of triangle A299778.

Views

Author

Keywords

Examples

Links

Crossrefs