A348705 a(n) is the total length of all line segments in the symmetric representation of sigma(n).

4, 8, 12, 16, 18, 24, 24, 32, 34, 40, 36, 48, 42, 54, 56, 64, 54, 72, 60, 80, 78, 82, 72, 96, 84, 96, 98, 112, 90, 120, 96, 128

Offset: 1

Author

Omar E. Pol, Oct 30 2021

Keywords

• nonn
• more

Comments

a(n) is also the number of toothpicks of length 1 needed to represent the symmetric representation of sigma(n) (see the examples).

The diagram is symmetric thus all terms are even.

If the symmetric representation of sigma(n) has only one part (cf. A174973) or if it has two parts and they meet at the center of the Dyck path (cf. A262259) then a(n) = 4*n, otherwise a(n) < 4*n. In other words: if n is a term of A279029 then a(n) = 4*n, otherwise a(n) < 4*n.

Examples

			Illustration of initial terms:
.                                                          _ _ _ _
.                                            _ _ _        |_ _ _  |_
.                                _ _ _      |_ _ _|             |   |_
.                      _ _      |_ _  |_          |_ _          |_ _  |
.              _ _    |_ _|_        |_  |           | |             | |
.        _    |_  |       | |         | |           | |             | |
.       |_|     |_|       |_|         |_|           |_|             |_|
.
n:       1      2        3          4           5               6
a(n):    4      8       12         16          18              24
.
.                                                  _ _ _ _ _
.                            _ _ _ _ _            |_ _ _ _ _|
.        _ _ _ _            |_ _ _ _  |                     |_ _
.       |_ _ _ _|                   | |_                    |_  |
.               |_                  |_  |_ _                  |_|_ _
.                 |_ _                |_ _  |                     | |
.                   | |                   | |                     | |
.                   | |                   | |                     | |
.                   | |                   | |                     | |
.                   |_|                   |_|                     |_|
.
n:              7                    8                      9
a(n):          24                   32                     34
.
Another way for the illustration of initial terms is as follows:
--------------------------------------------------------------------------
.  n  a(n)                             Diagram
--------------------------------------------------------------------------
            _
   1   4   |_|  _
              _| |  _
   2   8     |_ _| | |  _
                _ _|_| | |  _
   3  12       |_ _|  _| | | |  _
                  _ _|  _| | | | |  _
   4  16         |_ _ _|  _|_| | | | |  _
                    _ _ _|  _ _| | | | | |  _
   5  18           |_ _ _| |    _| | | | | | |  _
                      _ _ _|  _|  _|_| | | | | | |  _
   6  24             |_ _ _ _|  _|  _ _| | | | | | | |  _
                        _ _ _ _|  _|  _ _| | | | | | | | |  _
   7  24               |_ _ _ _| |  _|  _ _|_| | | | | | | | |  _
                          _ _ _ _| |  _| |  _ _| | | | | | | | | |  _
   8  32                 |_ _ _ _ _| |_ _| |  _ _| | | | | | | | | | |  _
                            _ _ _ _ _|  _ _|_|  _ _|_| | | | | | | | | | |
   9  34                   |_ _ _ _ _| |  _|  _|  _ _ _| | | | | | | | | |
                              _ _ _ _ _| |  _|  _|    _ _| | | | | | | | |
  10  40                     |_ _ _ _ _ _| |  _|     |  _ _|_| | | | | | |
                                _ _ _ _ _ _| |      _| |  _ _ _| | | | | |
  11  36                       |_ _ _ _ _ _| |  _ _|  _| |  _ _ _| | | | |
                                  _ _ _ _ _ _| |  _ _|  _|_|  _ _ _|_| | |
  12  48                         |_ _ _ _ _ _ _| |  _ _|  _ _| |  _ _ _| |
                                    _ _ _ _ _ _ _| |  _| |    _| |  _ _ _|
  13  42                           |_ _ _ _ _ _ _| | |  _|  _|  _| |
                                      _ _ _ _ _ _ _| | |_ _|  _|  _|
  14  54                             |_ _ _ _ _ _ _ _| |  _ _|  _|
                                        _ _ _ _ _ _ _ _| |  _ _|
  15  56                               |_ _ _ _ _ _ _ _| | |
                                          _ _ _ _ _ _ _ _| |
  16  64                                 |_ _ _ _ _ _ _ _ _|
...
		

Crossrefs

Cf. A008586 (upper bounds).

Cf. A237271 (number of parts or regions).

Cf. A340833 (number of vertices).

Cf. A340846 (number of edges).

Cf. A239931-A239934 (illustration of first 32 diagrams).

Cf. A000203, A139250, A174973, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A238443, A239660, A245092, A262259, A262626, A279029, A279228, A348854.

Formula

a(n) = 2*A348854(n).

a(n) = A008586(n) - A279228(n). - Omar E. Pol, Dec 13 2021