A211985 A list of certain compositions which arise from the ordered partitions of the positive integers in which the shells of each integer are arranged as a spiral.

1, 2, 1, 1, 3, 1, 1, 1, 1, 2, 4, 2, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 5, 2, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 4, 6, 3, 3, 4, 2, 2, 2, 2, 1, 4, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 3, 1, 5, 1, 7, 3, 4, 2, 5, 2, 2, 3, 1, 5, 1, 1, 2, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 2, 2, 2, 1, 4, 2, 1, 3, 3, 1, 6

Offset

1,2

Comments

In order to construct this sequence we use the following rules:

- Consider the partitions of positive integers.

- For each positive integer its shells must be arranged in a spiral.

- The sequence lists one spiral for each positive integer.

- If the integer j is odd then we use the same spiral of A211995.

- If the integer j is even then the first composition listed of each spiral is j.

Links

Table of n, a(n) for n=1..131.

Example

--------------------------------------------
.               Expanded        Geometric
Compositions   arrangement        model
--------------------------------------------
1;                 1;             |*|
--------------------------------------------
2;                 . 2;           |* *|
1,1;               1,1;           |o|*|
--------------------------------------------
3;               3 . .;         |* * *|
1,1,1;           1,1,1;         |*|o|o|
1,2;             1,. 2;         |*|o o|
--------------------------------------------
4,;              . . . 4;       |* * * *|
2,2;             . 2,. 2;       |* *|* *|
1,2,1;           1,. 2,1;       |o|o o|*|
1,1,1,1,;        1,1,1,1;       |o|o|o|*|
3,1;             3 . .,1;       |o o o|*|
--------------------------------------------
5;             5 . . . .;     |* * * * *|
2,3;           2 .,3 . .;     |* *|* * *|
1,3,1;         1,3 . .,1;     |*|o o o|o|
1,1,1,1,1;     1,1,1,1,1;     |*|o|o|o|o|
1,1,2,1;       1,1,. 2,1;     |*|o|o o|o|
1,2,2;         1,. 2,. 2;     |*|o o|o o|
1,4;           1,. . . 4;     |*|o o o o|
--------------------------------------------
6;             . . . . . 6;   |* * * * * *|
3,3;           . . 3,. . 3;   |* * *|* * *|
4,2;           . . . 4,. 2;   |* * * *|* *|
2,2,2;         . 2,. 2,. 2;   |* *|* *|* *|
1,4,1;         1,. . . 4,1;   |o|o o o o|*|
1,2,2,1;       1,. 2,. 2,1;   |o|o o|o o|*|
1,1,2,1,1;     1,1,. 2,1,1;   |o|o|o o|o|*|
1,1,1,1,1,1;   1,1,1,1,1,1;   |o|o|o|o|o|*|
1,3,1,1;       1,3 . .,1,1;   |o|o o o|o|*|
2,3,1;         2 .,3 . .,1;   |o o|o o o|*|
5,1;           5 . . . .,1;   |o o o o o|*|
--------------------------------------------
Note that * is a unitary element of every part of the last section of j.

Crossrefs

Rows sums give A036042, n>=1.

Mirror of A211986. Other spiral versions are A211987, A211988, A211995-A211998. See also A026792, A211983, A211984, A211989, A211992, A211993, A211994, A211999.

Cf. A000041, A026905, A135010, A138121, A138137, A138879, A182703, A206437.

Sequence in context: A352780 A331368 A106177 * A211009 A337266 A211986

Adjacent sequences: A211982 A211983 A211984 * A211986 A211987 A211988

Keyword

nonn,tabf

Author

Omar E. Pol, Aug 19 2012

Status

approved