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

Author

Omar E. Pol, Jan 24 2021

Keywords

• nonn

Comments

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).

Examples

			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                                 |_ _ _ _ _ _ _ _ _|
...
		

Crossrefs

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.

Formula

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

Extensions

More terms from Omar E. Pol, Oct 28 2021