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A211987 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. 2
1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 4, 2, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 4, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 2, 3, 5, 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, 1, 6, 1, 3, 3, 1, 4, 2, 1 (list; graph; refs; listen; history; text; internal format)
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 the last composition listed of each spiral is j.

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

This sequence represents a three-dimensional structure in which each column contains parts of the same size.

LINKS

Table of n, a(n) for n=1..86.

Omar E. Pol, Illustration of the transformation of the partitions of 7 (fig. A) into the rows 30..44 (fig. D)

EXAMPLE

----------------------------------------------

.                Expanded        Geometric

Compositions    arrangement        model

----------------------------------------------

1;                  1;              |*|

----------------------------------------------

2;                  . 2;            |* *|

1,1;                1,1;            |o|*|

----------------------------------------------

1,2;              1,. 2;          |*|o o|

1,1,1;            1,1,1;          |*|o|o|

3;                3 . .;          |* * *|

----------------------------------------------

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|*|

----------------------------------------------

1,4;            1,. . . 4;      |*|o o o o|

1,2,2;          1,. 2,. 2;      |*|o o|o o|

1,1,2,1;        1,1,. 2,1;      |*|o|o o|o|

1,1,1,1,1;      1,1,1,1,1;      |*|o|o|o|o|

1,3,1;          1,3 . .,1;      |*|o o o|o|

2,3;            2 .,3 . .;      |* *|* * *|

5;              5 . . . .;      |* * * * *|

----------------------------------------------

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 an unitary element of every part of the last section of j.

CROSSREFS

Rows sums give A036042, n>=1.

Mirror of A211988. Other spiral versions are A211985, A211986, A211995-A211998. See also A026792, A211983, A211984, A211989, A211992, A211993, A211994, A211999.

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

Sequence in context: A170981 A161096 A214247 * A165983 A300719 A083894

Adjacent sequences:  A211984 A211985 A211986 * A211988 A211989 A211990

KEYWORD

nonn,tabf

AUTHOR

Omar E. Pol, Aug 18 2012

STATUS

approved

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Last modified July 4 12:18 EDT 2020. Contains 335448 sequences. (Running on oeis4.)