A279228 Number of unit steps that are shared by the smallest and largest Dyck path of the symmetric representation of sigma(n).

0, 0, 0, 0, 2, 0, 4, 0, 2, 0, 8, 0, 10, 2, 4, 0, 14, 0, 16, 0, 6, 6, 20, 0, 16, 8, 10, 0, 26, 0, 28, 0

Offset: 1

Author

Omar E. Pol, Dec 08 2016

Keywords

• nonn
• more

Comments

a(n) is also the number of unit steps that are shared by the largest Dyck path of the symmetric representation of sigma(n) and the largest Dyck path of the symmetric representation of sigma(n-1), in a quadrant of the square grid.

For more information about the Dyck paths of the symmetric representation of sigma(n) see A237593.

Examples

			Illustration of initial terms (n = 1..12) using the spiral described in A239660:
.               _ _ _ _ _ _
.              |  _ _ _ _ _|_ _ _ _ _
.         0   _| |         |_ _ _ _ _|
.           _|_ _|                   |_ _ 2
.       _ _| |      _ _ _ _          |_  |
.      |  _ _|  0 _|  _ _ _|_ _ _      |_|_ _
.      | |      _|   |     |_ _ _|  2      | |
.      | |     |  _ _|           |_ _      | |
.      | |     | |    0 _ _        | |     | |
.      | |     | |     |  _|_ 0    | |     | |
.     _|_|    _|_|    _|_| |_|    _|_|    _|_|    _
.    | |     | |     | |         | |     | |     | |
.    | |     | |     |_|_ _     _| |     | |     | |
.    | |     | |      0|_ _|_ _|  _|     | |     | |
.    | |     |_|_          |_ _ _|0   _ _| |     | |
.    | |         |_                 _|  _ _|     | |
.    |_|_ _     4  |_ _ _ _        |  _|    _ _ _| |
.          |_      |_ _ _ _|_ _ _ _| |  0 _|    _ _|
.            |_            |_ _ _ _ _|  _|     |
.         8    |                       |      _|
.              |_ _ _ _ _ _            |  _ _|
.              |_ _ _ _ _ _|_ _ _ _ _ _| |      0
.                          |_ _ _ _ _ _ _|
.
.
For an illustration of the following examples see the last lap of the above spiral starting in the first quadrant.
For n = 9 the Dyck paths of the symmetric representation of sigma(9) share 2 unit steps, so a(9) = 2.
For n = 10 the Dyck paths of the symmetric representation of sigma(10) meet at the center, but they do not share unit steps, so a(10) = 0.
For n = 11 the Dyck paths of the symmetric representation of sigma(11) share 8 unit steps, so a(11) = 8.
For n = 12 the Dyck paths of the symmetric representation of sigma(12) do not share unit steps, so a(12) = 0.
Note that we can find the spiral on the terraces of the stepped pyramid described in A244050.
		

Crossrefs

Cf. A279029 gives the indices of the zero values.

Cf. A279244 gives the indices of the positive values.

Cf. A000203, A005843, A008586, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239660, A239931-A239934, A244050, A244361, A244363, A245092, A262626, A348705.

Formula

a(n) = 2*n - A244363(n) = 2*(n - A244361(n)).

a(n) = A008586(n) - A348705(n). - Omar E. Pol, Dec 13 2021