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

1,1

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.

Links

Table of n, a(n) for n=1..32.

Formula

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

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

Example

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.

Sequence in context: A068306 A378040 A311118 * A272405 A311119 A321177

Adjacent sequences: A348702 A348703 A348704 * A348706 A348707 A348708

Keyword

nonn,more

Author

Omar E. Pol, Oct 30 2021

Status

approved