A211988 -id:A211988 - cp's OEIS frontend

A211999 A list of ordered partitions of the positive integers in which the shells of each integer are assembled by their tails.

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

Offset: 1

Omar E. Pol, Aug 14 2012

The sequence lists the partitions of all positive integers. Each row of the irregular array is a partition of j.
At stage 1, we start with 1.
At stage j > 1, we write the partitions of j using the following rules:
First we copy the last A000041(j-1) rows of the array in descending order, as a mirror image, starting with the row that contains the part of size j-1. At the end of each new row, we added a part of size 1.
Second, we write the partitions of j that do not contain 1 as a part, in reverse-lexicographic order, such that the last row (or partition of j) is j.
Note that the table can be partially folded. In this case we have a three-dimensional structure in which each column contains parts of the same size (see example). Also the table can be completely folded, therefore stacked parts have the same size.

			A table of partitions.
---------------------------------------------------------
.              Expanded       Geometric  Side view of the
Partitions     version        model      folded table
---------------------------------------------------------
1;             1;             |*|                /
---------------------------------------------------------
1,1;           1,1;           |o|*|              \
2;             . 2;           |* *|               \
---------------------------------------------------------
2,1;           . 2,1;         |o o|*|             /
1,1,1;         1,1,1;         |o|o|*|            /
3;             . . 3;         |* * *|           /
---------------------------------------------------------
3,1;           . . 3,1;       |o o o|*|         \
1,1,1,1;       1,1,1,1;       |o|o|o|*|          \
2,1,1;         . 2,1,1;       |o o|o|*|           \
2,2;           . 2,. 2;       |* *|* *|            \
4;             . . . 4;       |* * * *|             \
---------------------------------------------------------
4,1;           . . . 4,1;     |o o o o|*|           /
2,2,1;         . 2,. 2,1;     |o o|o o|*|          /
2,1,1,1;       . 2,1,1,1;     |o o|o|o|*|         /
1,1,1,1,1;     1,1,1,1,1;     |o|o|o|o|*|        /
3,1,1;         . . 3,1,1;     |o o o|o|*|       /
3,2;           . . 3,. 2;     |* * *|* *|      /
5;             . . . . 5;     |* * * * *|     /
---------------------------------------------------------
5,1;           . . . . 5,1;   |o o o o o|*|   \
3,2,1;         . . 3,. 2,1;   |o o o|o o|*|    \
3,1,1,1;       . . 3,1,1,1;   |o o o|o|o|*|     \
1,1,1,1,1,1;   1,1,1,1,1,1;   |o|o|o|o|o|*|      \
2,1,1,1,1;     . 2,1,1,1,1;   |o o|o|o|o|*|       \
2,2,1,1;       . 2,. 2,1,1;   |o o|o o|o|*|        \
4,1,1;         . . . 4,1,1;   |o o o o|o|*|         \
2,2,2;         . 2, .2,. 2;   |* *|* *|* *|          \
4,2;           . . . 4,. 2;   |* * * *|* *|           \
3,3;           . . 3,. . 3;   |* * *|* * *|            \
6;             . . . . . 6;   |* * * * * *|             \
---------------------------------------------------------
Note that * is a unitary element of every part of the last section of j.

Omar E. Pol, Illustration of initial terms

Rows sums give A036042, n>=1.
Other versions are A211983, A211984, A211989. See also A026792, A211992-A211994. Spiral arrangements are A211985-A211988, A211995-A211998.
Cf. A000041, A026905, A135010, A138121, A138137, A138879, A182703, A206437.

A211983 A list of ordered partitions of the positive integers in which the shells of each integer are assembled by their tails.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 19 2012

The order of the partitions of the odd integers is the same as A211999. The order of the partitions of the even integers is the same as A211989.

			A table of partitions.
--------------------------------------------
.              Expanded       Geometric
Partitions     arrangement    model
--------------------------------------------
1;             1;             |*|
--------------------------------------------
2;             . 2;           |* *|
1,1;           1,1;           |o|*|
--------------------------------------------
2,1;           . 2,1;         |o o|*|
1,1,1;         1,1,1;         |o|o|*|
3;             . . 3;         |* * *|
--------------------------------------------
4;             . . . 4;       |* * * *|
2,2;           . 2,. 2;       |* *|* *|
2,1,1;         . 2,1,1;       |o o|o|*|
1,1,1,1;       1,1,1,1;       |o|o|o|*|
3,1;           . . 3,1;       |o o o|*|
--------------------------------------------
4,1;           . . . 4,1;     |o o o o|*|
2,2,1;         . 2,. 2,1;     |o o|o o|*|
2,1,1,1;       . 2,1,1,1;     |o o|o|o|*|
1,1,1,1,1;     1,1,1,1,1;     |o|o|o|o|*|
3,1,1;         . . 3,1,1;     |o o o|o|*|
3,2;           . . 3,. 2;     |* * *|* *|
5;             . . . . 5;     |* * * * *|
--------------------------------------------
6;             . . . . . 6;   |* * * * * *|
3,3;           . . 3,. . 3;   |* * *|* * *|
4,2;           . . . 4,. 2;   |* * * *|* *|
2,2,2;         . 2,. 2,. 2;   |* *|* *|* *|
4,1,1;         . . . 4,1,1;   |o o o o|o|*|
2,2,1,1;       . 2,. 2,1,1;   |o o|o o|o|*|
2,1,1,1,1;     . 2,1,1,1,1;   |o o|o|o|o|*|
1,1,1,1,1,1;   1,1,1,1,1,1;   |o|o|o|o|o|*|
3,1,1,1;       . . 3,1,1,1;   |o o o|o|o|*|
3,2,1;         . . 3,. 2,1;   |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.

Rows sums give A036042, n>=1.
Other versions are A211984, A211989, A211999. See also A026792, A211992-A211994. Spiral arrangements are A211985-A211988, A211995-A211998.
Cf. A000041, A026905, A135010, A138121, A138137, A138879, A182703, A206437.

A211984 A list of ordered partitions of the positive integers in which the shells of each integer are assembled by their tails.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 19 2012

The order of the partitions of the odd integers is the same as A211989. The order of the partitions of the even integers is the same as A211999.

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

Rows sums give A036042, n>=1.
Other versions are A211983, A211989, A211999. See also A026792, A211992-A211994. Spiral arrangements are A211985-A211988, A211995-A211998.
Cf. A000041, A026905, A135010, A138121, A138137, A138879, A182703, A206437.

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.

Original entry on oeis.org

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

Omar E. Pol, Aug 19 2012

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.

			--------------------------------------------
.               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.

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.

A211989 A list of ordered partitions of the positive integers in which the shells of each integer are assembled by their tails.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 18 2012

The sequence lists the partitions of all positive integers. Each row of the irregular array is a partition of j.
At stage 1, we start with 1.
At stage j > 1, we write the partitions of j using the following rules:
First, we write the partitions of j that do not contain 1 as a part, in reverse-lexicographic order, starting with the partition that contains the part of size j.
Second, we copy from this array the partitions of j-1 in descending order, as a mirror image, starting with the partition that contains the part of size j-2 together with the part of size 1. At the end of each new row, we added a part of size 1.

			A table of partitions.
--------------------------------------------
.              Expanded       Geometric
Partitions     arrangement    model
--------------------------------------------
1;             1;             |*|
--------------------------------------------
2;             . 2;           |* *|
1,1;           1,1;           |o|*|
--------------------------------------------
3;             . . 3;         |* * *|
1,1,1;         1,1,1;         |o|o|*|
2,1;           . 2,1;         |o o|*|
--------------------------------------------
4;             . . . 4;       |* * * *|
2,2;           . 2,. 2;       |* *|* *|
2,1,1;         . 2,1,1;       |o o|o|*|
1,1,1,1;       1,1,1,1;       |o|o|o|*|
3,1;           . . 3,1;       |o o o|*|
--------------------------------------------
5;             . . . . 5;     |* * * * *|
3,2;           . . 3,. 2;     |* * *|* *|
3,1,1;         . . 3,1,1;     |o o o|o|*|
1,1,1,1,1;     1,1,1,1,1;     |o|o|o|o|*|
2,1,1,1;       . 2,1,1,1;     |o o|o|o|*|
2,2,1;         . 2,. 2,1;     |o o|o o|*|
4,1;           . . . 4,1;     |o o o o|*|
--------------------------------------------
6;             . . . . . 6;   |* * * * * *|
3,3;           . . 3,. . 3;   |* * *|* * *|
4,2;           . . . 4,. 2;   |* * * *|* *|
2,2,2;         . 2,. 2,. 2;   |* *|* *|* *|
4,1,1;         . . . 4,1,1;   |o o o o|o|*|
2,2,1,1;       . 2,. 2,1,1;   |o o|o o|o|*|
2,1,1,1,1;     . 2,1,1,1,1;   |o o|o|o|o|*|
1,1,1,1,1,1;   1,1,1,1,1,1;   |o|o|o|o|o|*|
3,1,1,1;       . . 3,1,1,1;   |o o o|o|o|*|
3,2,1;         . . 3,. 2,1;   |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.

Rows sums give A036042, n>=1.
Other versions are A211983, A211984, A211999. See also A026792, A211992-A211994. Spiral arrangements are A211985-A211988, A211995-A211998.
Cf. A000041, A026905, A135010, A138121, A138137, A138879, A182703, A206437.

A211994 A list of ordered partitions of the positive integers.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 18 2012

The order of the partitions of the odd integers is the same as A026792. The order of the partitions of the even integers is the same as A211992.

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

Rows sums give A036042, n>=1.
Other versions are A026792, A211992, A211993. See also A211983, A211984, A211989, A211999. Spiral arrangements are A211985-A211988, A211995-A211998.
Cf. A000041, A026905, A135010, A138121, A138137, A138879, A182703, A206437.

A217548 The Berndt-type sequences number 7 for the argument 2*Pi/13.

Original entry on oeis.org

6, 7, -65, -295, -1303, 20631, 89967, 392616, -6178549, -26970688, -117731275, 1852943703, 8088348131, 35306734632, -555682818080, -2425630962790, -10588208505263, 166644858132571, -727427431532172, 3175319503526856, -49975467287014789

Offset: 0

Roman Witula, Oct 06 2012

sign

a(n) is the rational component (with respect to the field Q(sqrt(13))) of the number A(2*n)*2*13^(floor((n+1)/3)/2), where A(n) = sqrt((13-3*sqrt(13))/2)*A(n-1) + (sqrt(13)-3)*A(n-2)/2 - sqrt((13-3*sqrt(13))/26)*A(n-3), with A(-1) = sqrt((13-3*sqrt(13))/2), A(0)=3, and A(1) = sqrt((13-3*sqrt(13))/2).
The basic sequence A(n) is defined  by the relation A(n) = s(1)^(-n) + s(3)^(-n) + s(9)^(-n), where s(j) = 2*sin(2*Pi*j/13). The sequence with respective positive powers is discussed in A216508 (see sequence Y(n) in Comments to A216508).
We note that s(1) + s(3) + s(9) = s(1)^(-1) + s(3)^(-1) + s(9)^(-1) = sqrt((13-3*sqrt(13))/2).
The numbers of other Berndt-type sequences for the argument 2*Pi/13 in crossrefs are given.

R. Witula and D. Slota, Quasi-Fibonacci numbers of order 13, Thirteenth International Conference on Fibonacci Numbers and their Applications, Congressus Numerantium, 201 (2010), 89-107.
R. Witula, On some applications of formulas for sums of the unimodular complex numbers, Wyd. Pracowni Komputerowej Jacka Skalmierskiego, Gliwice 2011 (in Polish).

R. Witula and D. Slota, Quasi-Fibonacci numbers of order 13, (abstract) see p. 15.

Cf. A019698, A216605, A216486, A216508, A216597, A216540, A161905, A216450, A216801, A216861, A217548, A217549, A211988.

A211993 A list of ordered partitions of the positive integers.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 18 2012

The order of the partitions of the odd integers is the same as A211992. The order of the partitions of the even integers is the same as A026792.

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

Rows sums give A036042, n>=1.
Other versions are A026792, A211992, A211994. See also A211983, A211984, A211989, A211999. Spiral arrangements are A211985-A211988, A211995-A211998.
Cf. A000041, A026905, A135010, A138121, A138137, A138879, A182703, A206437.

A211986 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 the arms of a spiral.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 19 2012

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 as the arms of a spiral.
- The sequence lists one spiral for each positive integer.
- If the integer j is odd then the first composition listed of each spiral is j.
- If the integer j is even then we use the same spiral of A211988.

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

Omar E. Pol, Illustration of the rows 30..44 (figure D)

Rows sums give A036042, n>=1.
Mirror of A211985. 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.

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.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 18 2012

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.

			----------------------------------------------
.                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 a unitary element of every part of the last section of j.

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

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.

cp's OEIS Frontend

A211999 A list of ordered partitions of the positive integers in which the shells of each integer are assembled by their tails.

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A211983 A list of ordered partitions of the positive integers in which the shells of each integer are assembled by their tails.

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A211984 A list of ordered partitions of the positive integers in which the shells of each integer are assembled by their tails.

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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.

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A211989 A list of ordered partitions of the positive integers in which the shells of each integer are assembled by their tails.

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A211994 A list of ordered partitions of the positive integers.

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A217548 The Berndt-type sequences number 7 for the argument 2*Pi/13.

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A211993 A list of ordered partitions of the positive integers.

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A211986 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 the arms of a spiral.

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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.

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