A182703 -id:A182703 - cp's OEIS frontend

A211984 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, 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.

A340031 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of A000041(m-1) copies of the j-th row of triangle A127093, where j = n - m + 1 and 1 <= m <= n.

Original entry on oeis.org

1, 1, 2, 1, 1, 0, 3, 1, 2, 1, 1, 1, 2, 0, 4, 1, 0, 3, 1, 2, 1, 2, 1, 1, 1, 1, 0, 0, 0, 5, 1, 2, 0, 4, 1, 0, 3, 1, 0, 3, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 3, 0, 0, 6, 1, 0, 0, 0, 5, 1, 2, 0, 4, 1, 2, 0, 4, 1, 0, 3, 1, 0, 3, 1, 0, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Omar E. Pol, Dec 26 2020

Another version of A338156 which is the main sequence with further information about the correspondence divisor/part.

			Triangle begins:
[1];
[1,2],      [1];
[1,0,3],    [1,2],    [1],    [1];
[1,2,0,4],  [1,0,3],  [1,2],  [1,2],  [1],  [1],  [1];
[1,0,0,0,5],[1,2,0,4],[1,0,3],[1,0,3],[1,2],[1,2],[1,2],[1],[1],[1],[1],[1];
[...
Written as an irregular tetrahedron the first five slices are:
[1],
-------
[1, 2],
[1],
----------
[1, 0, 3],
[1, 2],
[1],
[1];
-------------
[1, 2, 0, 4],
[1, 0, 3],
[1, 2],
[1, 2],
[1],
[1],
[1];
----------------
[1, 0, 0, 0, 5],
[1, 2, 0, 4],
[1, 0, 3],
[1, 0, 3],
[1, 2],
[1, 2],
[1, 2],
[1],
[1],
[1],
[1],
[1];
.
The following table formed by three zones shows the correspondence between divisors and parts (n = 1..5):
.
|---|---------|-----|-------|---------|-----------|-------------|
| n |         |  1  |   2   |    3    |     4     |      5      |
|---|---------|-----|-------|---------|-----------|-------------|
| P |         |     |       |         |           |             |
| A |         |     |       |         |           |             |
| R |         |     |       |         |           |             |
| T |         |     |       |         |           |  5          |
| I |         |     |       |         |           |  3 2        |
| T |         |     |       |         |  4        |  4 1        |
| I |         |     |       |         |  2 2      |  2 2 1      |
| O |         |     |       |  3      |  3 1      |  3 1 1      |
| N |         |     |  2    |  2 1    |  2 1 1    |  2 1 1 1    |
| S |         |  1  |  1 1  |  1 1 1  |  1 1 1 1  |  1 1 1 1 1  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A181187 |  1  |  3 1  |  6 2 1  | 12 5 2 1  | 20 8 4 2 1  |
| L |         |  |  |  |/|  |  |/|/|  |  |/|/|/|  |  |/|/|/|/|  |
| I | A066633 |  1  |  2 1  |  4 1 1  |  7 3 1 1  | 12 4 2 1 1  |
| N |         |  *  |  * *  |  * * *  |  * * * *  |  * * * * *  |
| K | A002260 |  1  |  1 2  |  1 2 3  |  1 2 3 4  |  1 2 3 4 5  |
|   |         |  =  |  = =  |  = = =  |  = = = =  |  = = = = =  |
|   | A138785 |  1  |  2 2  |  4 2 3  |  7 6 3 4  | 12 8 6 4 5  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A127093 |  1  |  1 2  |  1 0 3  |  1 2 0 4  |  1 0 0 0 5  |
|   |---------|-----|-------|---------|-----------|-------------|
|   | A127093 |     |  1    |  1 2    |  1 0 3    |  1 2 0 4    |
|   |---------|-----|-------|---------|-----------|-------------|
| D | A127093 |     |       |  1      |  1 2      |  1 0 3      |
| I | A127093 |     |       |  1      |  1 2      |  1 0 3      |
| V |---------|-----|-------|---------|-----------|-------------|
| I | A127093 |     |       |         |  1        |  1 2        |
| S | A127093 |     |       |         |  1        |  1 2        |
| O | A127093 |     |       |         |  1        |  1 2        |
| R |---------|-----|-------|---------|-----------|-------------|
| S | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|---|---------|-----|-------|---------|-----------|-------------|
.
The table is essentially the same table of A338156 but here, in the lower zone, every row is A127093 instead of A027750.
.

Paolo Xausa, Table of n, a(n) for n = 1..11552 (rows 1..17 of the triangle, flattened)

Row sums give A066186.
Nonzero terms gives A338156.
Cf. A000070, A000041, A002260, A026792, A027750, A058399, A066633, A127093, A135010, A138121, A138785, A176206, A181187, A182703, A207031, A207383, A211992, A221529, A221530, A221531, A221649, A221650, A237593, A245095, A302246, A302247, A336811, A337209,  A339106, A339258, A339278, A339304, A340011, A340032, A340035, A340061.

A127093row[n_]:=Table[Boole[Divisible[n,k]]k,{k,n}];
A340031row[n_]:=Flatten[Table[ConstantArray[A127093row[n-m+1],PartitionsP[m-1]],{m,n}]];
Array[A340031row,7] (* Paolo Xausa, Sep 28 2023 *)

A182713 Number of 3's in the last section of the set of partitions of n.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 2, 3, 6, 6, 10, 14, 18, 24, 35, 42, 58, 76, 97, 124, 164, 202, 261, 329, 412, 514, 649, 795, 992, 1223, 1503, 1839, 2262, 2741, 3346, 4056, 4908, 5919, 7150, 8568, 10297, 12320, 14721, 17542, 20911, 24808, 29456, 34870, 41232, 48652, 57389

Offset: 1

Omar E. Pol, Nov 28 2010

nonn

Also number of 3's in all partitions of n that do not contain 1 as a part.
Also 0 together with the first differences of A024787. - Omar E. Pol, Nov 13 2011
a(n) is the number of partitions of n having fewer 1s than 2s; e.g., a(7) counts these 3 partitions: [5, 2], [3, 2, 2], [2, 2, 2, 1]. - Clark Kimberling, Mar 31 2014
The last section of the set of partitions of n is also the n-th section of the set of partitions of any integer >= n. - Omar E. Pol, Apr 07 2014

			a(7) = 2 counts the 3's in 7 = 4+3 = 3+2+2. The 3's in 7 = 3+3+1 = 3+2+1+1 = 3+1+1+1+1 do not count.
From _Omar E. Pol_, Oct 27 2012: (Start)
--------------------------------------
Last section                   Number
of the set of                    of
partitions of 7                 3's
--------------------------------------
7 .............................. 0
4 + 3 .......................... 1
5 + 2 .......................... 0
3 + 2 + 2 ...................... 1
.   1 .......................... 0
.       1 ...................... 0
.       1 ...................... 0
.           1 .................. 0
.       1 ...................... 0
.           1 .................. 0
.           1 .................. 0
.               1 .............. 0
.               1 .............. 0
.                   1 .......... 0
.                       1 ...... 0
------------------------------------
.       5 - 3 =                  2
.
In the last section of the set of partitions of 7 the difference between the sum of the third column and the sum of the fourth column is 5 - 3 = 2 equaling the number of 3's, so a(7) = 2 (see also A024787).
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Column 3 of A194812.

Cf. A135010, A138121, A174455, A182703, A182712, A182714, A240056.

b:= proc(n, i) option remember; local g, h;
      if n=0 then [1, 0]
    elif i<2 then [0, 0]
    else g:= b(n, i-1); h:= `if`(i>n, [0, 0], b(n-i, i));
         [g[1]+h[1], g[2]+h[2]+`if`(i=3, h[1], 0)]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..70);  # Alois P. Heinz, Mar 18 2012

z = 60; f[n_] := f[n] = IntegerPartitions[n]; t1 = Table[Count[f[n], p_ /; Count[p, 1] < Count[p, 2]], {n, 0, z}] (* Clark Kimberling, Mar 31 2014 *)
b[n_, i_] := b[n, i] = Module[{g, h}, If[n == 0, {1, 0}, If[i<2, {0, 0}, g = b[n, i-1]; h = If[i>n, {0, 0}, b[n-i, i]]; Join[g[[1]] + h[[1]], g[[2]] + h[[2]] + If[i == 3, h[[1]], 0]]]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 70}] (* Jean-François Alcover, Nov 30 2015, after Alois P. Heinz *)
Table[Count[Flatten@Cases[IntegerPartitions[n], x_ /; Last[x] != 1], 3], {n, 51}] (* Robert Price, May 15 2020 *)

A182713 = lambda n: sum(list(p).count(3) for p in Partitions(n) if 1 not in p) # D. S. McNeil, Nov 29 2010

It appears that A000041(n) = a(n+1) + a(n+2) + a(n+3), n >= 0. - Omar E. Pol, Feb 04 2012

a(n) ~ A000041(n)/3 ~ exp(Pi*sqrt(2*n/3)) / (12*sqrt(3)*n). - Vaclav Kotesovec, Jan 03 2019

A207034 Sum of all parts minus the number of parts of the n-th partition in the list of colexicographically ordered partitions of j, if 1<=n<=A000041(j).

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 3, 4, 4, 5, 4, 5, 5, 6, 4, 5, 5, 6, 6, 6, 7, 5, 6, 6, 7, 6, 7, 7, 8, 5, 6, 6, 7, 7, 7, 8, 7, 8, 8, 8, 9, 6, 7, 7, 8, 7, 8, 8, 9, 8, 8, 9, 9, 9, 10, 6, 7, 7, 8, 8, 8, 9, 8, 9, 9, 9, 10, 8, 9, 9, 10, 9, 10, 10, 10, 11, 7, 8, 8, 9, 8, 9

Offset: 1

Omar E. Pol, Feb 20 2012

a(n) is also the column number in which is located the part of size 1 in the n-th zone of the tail of the last section of the set of partitions of k in colexicographic order, minus the column number in which is located the part of size 1 in the first row of the same tail, when k -> infinity (see example). For the definition of "section" see A135010.

			Illustration of initial terms, n = 1..15. Consider the last 15 rows of the tail of the last section of the set of partitions in colexicographic order of any integer >= 8. The tail contains at least A000041(8-1) = 15 parts of size 1. a(n) is also the number of dots in the n-th row of the diagram.
----------------------------------
n      Tail                  a(n)
----------------------------------
15        1 . . . . . .       6
14          1 . . . . .       5
13          1 . . . . .       5
12            1 . . . .       4
11          1 . . . . .       5
10            1 . . . .       4
9             1 . . . .       4
8               1 . . .       3
7             1 . . . .       4
6               1 . . .       3
5               1 . . .       3
4                 1 . .       2
3                 1 . .       2
2                   1 .       1
1                     1       0
----------------------------------
Written as a triangle:
0;
1;
2;
2,3;
3,4;
3,4,4,5;
4,5,5,6;
4,5,5,6,6,6,7;
5,6,6,7,6,7,7,8;
5,6,6,7,7,7,8,7,8,8,8,9;
6,7,7,8,7,8,8,9,8,8,9,9,9,10;
6,7,7,8,8,8,9,8,9,9,9,10,8,9,9,10,9,10,10,10,11;
...
Consider a matrix [j X A000041(j)] in which the rows represent the partitions of j in colexicographic order (see A211992). Every part of every partition is located in a cell of the matrix. We can see that a(n) is the number of empty cells in row n for any integer j, if A000041(j) >= n. The number of empty cells in row n equals the sum of all parts minus the number of parts in the n-th partition of j.
Illustration of initial terms. The smallest part of every partition is located in the last column of the matrix.
---------------------------------------------------------
.   j: 1    2       3         4           5             6
n a(n)
---------------------------------------------------------
1  0 | 1  1 1   1 1 1   1 1 1 1   1 1 1 1 1   1 1 1 1 1 1
2  1 |    . 2   . 2 1   . 2 1 1   . 2 1 1 1   . 2 1 1 1 1
3  2 |          . . 3   . . 3 1   . . 3 1 1   . . 3 1 1 1
4  2 |                  . . 2 2   . . 2 2 1   . . 2 2 1 1
5  3 |                  . . . 4   . . . 4 1   . . . 4 1 1
6  3 |                            . . . 3 2   . . . 3 2 1
7  4 |                            . . . . 5   . . . . 5 1
8  3 |                                        . . . 2 2 2
9  4 |                                        . . . . 4 2
10 4 |                                        . . . . 3 3
11 5 |                                        . . . . . 6
...
Illustration of initial terms. In this case the largest part of every partition is located in the first column of the matrix.
---------------------------------------------------------
.   j: 1    2       3         4           5             6
n a(n)
---------------------------------------------------------
1  0 | 1  1 1   1 1 1   1 1 1 1   1 1 1 1 1   1 1 1 1 1 1
2  1 |    2 .   2 1 .   2 1 1 .   2 1 1 1 .   2 1 1 1 1 .
3  2 |          3 . .   3 1 . .   3 1 1 . .   3 1 1 1 . .
4  2 |                  2 2 . .   2 2 1 . .   2 2 1 1 . .
5  3 |                  4 . . .   4 1 . . .   4 1 1 . . .
6  3 |                            3 2 . . .   3 2 1 . . .
7  4 |                            5 . . . .   5 1 . . . .
8  3 |                                        2 2 2 . . .
9  4 |                                        4 2 . . . .
10 4 |                                        3 3 . . . .
11 5 |                                        6 . . . . .
...

Alois P. Heinz, Table of n, a(n) for n = 1..10143

Row r has length A187219(r). Partial sums give A207038. Row sums give A207035. Right border gives A001477. Where records occur give A000041 without repetitions.

Cf. A135010, A138121, A141285, A182703, A194548, A196087, A207031, A207032, A207035, A211992, A228716, A230440.

a(n) = t(n) - A194548(n), if n >= 2, where t(n) is the n-th element of the following sequence: triangle read by rows in which row n lists n repeated k times, where k = A187219(n).
a(n) = A000120(A194602(n-1)) = A000120(A228354(n)-1).
a(n) = i - A193173(i,n), i >= 1, 1<=n<=A000041(i).

A210979 Total area of the shadows of the three views of the version "Tree" of the shell model of partitions with n shells.

Original entry on oeis.org

0, 3, 8, 15, 27, 42, 69, 102, 155, 225, 327, 458, 652, 894, 1232, 1669, 2257, 2999, 3996, 5242, 6877, 8928, 11564, 14845, 19045, 24223, 30756, 38815, 48877, 61195, 76496, 95124, 118067, 145930, 179991, 221160, 271268, 331538, 404463, 491948, 597253

Offset: 0

Omar E. Pol, Apr 28 2012

nonn

The physical model shows each part of a partition as an object, for example; a cube of side 1 which is labeled with the size of the part. Note that on the branches of the tree each column contains parts of the same size, as a periodic structure. For the large version of this model see A210980.

			For n = 7 the three views of the shell model of partitions version "tree" with seven shells looks like this:
.
.         A194805(7) = 25        A006128(7) = 54
.
.                        7       7
.                      4         4 3
.                    5           5 2
.                  3             3 2 2
.        6       1               6 1
.          3     1               3 3 1
.            4   1               4 2 1
.              2 1               2 2 2 1
.                1   5           5 1 1
.                1 3             3 2 1 1
.            4   1               4 1 1 1
.              2 1               2 2 1 1 1
.                1 3             3 1 1 1 1
.              2 1               2 1 1 1 1 1
.                1               1 1 1 1 1 1 1
-------------------------------------------------
.
.        6 3 4 2 1 3 5 4 7
.          3 2 2 1 2 2 3
.              2 1 2
.                1
.                1
.                1
.                1
.
.         A194803(7) = 23
.
The areas of the shadows of the three views are A006128(7) = 54, A194803(7) = 23 and A194805(7) = 25, therefore the total area of the three shadows is 54+23+25 = 102, so a(7) = 102.

Other versions: A207380, A210970, A210980, A210990, A210991.

Cf. A006128, A135010, A138121, A182703, A194803, A194805.

a(n) = A006128(n) + A194803(n) + A194805(n).

A340011 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of the j-th row of triangle A127093 but with every term multiplied by A000041(m-1), where j = n - m + 1 and 1 <= m <= n.

Original entry on oeis.org

1, 1, 2, 1, 1, 0, 3, 1, 2, 2, 1, 2, 0, 4, 1, 0, 3, 2, 4, 3, 1, 0, 0, 0, 5, 1, 2, 0, 4, 2, 0, 6, 3, 6, 5, 1, 2, 3, 0, 0, 6, 1, 0, 0, 0, 5, 2, 4, 0, 8, 3, 0, 9, 5, 10, 7, 1, 0, 0, 0, 0, 0, 7, 1, 2, 3, 0, 0, 6, 2, 0, 0, 0, 10, 3, 6, 0, 12, 5, 0, 15, 7, 14, 11, 1, 2, 0, 4, 0, 0, 0, 8

Offset: 1

Omar E. Pol, Dec 26 2020

This triangle is a condensed version of the more irregular triangle A340031.

For further information about the correspondence divisor/part see A338156.

			Triangle begins:
[1];
[1, 2],          [1];
[1, 0, 3],       [1, 2],       [2];
[1, 2, 0, 4],    [1, 0, 3],    [2, 4],    [3];
[1, 0, 0, 0, 5], [1, 2, 0, 4], [2, 0, 6], [3, 6], [5];
[...
Row sums give A066186.
Written as an irregular tetrahedron the first five slices are:
--
1;
-----
1, 2,
1;
--------
1, 0, 3,
1, 2,
2;
-----------
1, 2, 0, 4,
1, 0, 3,
2, 4,
3;
--------------
1, 0, 0, 0, 5,
1, 2, 0, 4,
2, 0, 6,
3, 6,
5;
--------------
Row sums give A339106.
The following table formed by four zones shows the correspondence between divisor and parts (n = 1..5):
.
|---|---------|-----|-------|---------|-----------|-------------|
| n |         |  1  |   2   |    3    |     4     |      5      |
|---|---------|-----|-------|---------|-----------|-------------|
| P |         |     |       |         |           |             |
| A |         |     |       |         |           |             |
| R |         |     |       |         |           |             |
| T |         |     |       |         |           |  5          |
| I |         |     |       |         |           |  3 2        |
| T |         |     |       |         |  4        |  4 1        |
| I |         |     |       |         |  2 2      |  2 2 1      |
| O |         |     |       |  3      |  3 1      |  3 1 1      |
| N |         |     |  2    |  2 1    |  2 1 1    |  2 1 1 1    |
| S |         |  1  |  1 1  |  1 1 1  |  1 1 1 1  |  1 1 1 1 1  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A181187 |  1  |  3 1  |  6 2 1  | 12 5 2 1  | 20 8 4 2 1  |
| L |         |  |  |  |/|  |  |/|/|  |  |/|/|/|  |  |/|/|/|/|  |
| I | A066633 |  1  |  2 1  |  4 1 1  |  7 3 1 1  | 12 4 2 1 1  |
| N |         |  *  |  * *  |  * * *  |  * * * *  |  * * * * *  |
| K | A002260 |  1  |  1 2  |  1 2 3  |  1 2 3 4  |  1 2 3 4 5  |
|   |         |  =  |  = =  |  = = =  |  = = = =  |  = = = = =  |
|   | A138785 |  1  |  2 2  |  4 2 3  |  7 6 3 4  | 12 8 6 4 5  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A127093 |  1  |  1 2  |  1 0 3  |  1 2 0 4  |  1 0 0 0 5  |
|   |---------|-----|-------|---------|-----------|-------------|
|   | A127093 |     |  1    |  1 2    |  1 0 3    |  1 2 0 4    |
|   |---------|-----|-------|---------|-----------|-------------|
| D | A127093 |     |       |  1      |  1 2      |  1 0 3      |
| I | A127093 |     |       |  1      |  1 2      |  1 0 3      |
| V |---------|-----|-------|---------|-----------|-------------|
| I | A127093 |     |       |         |  1        |  1 2        |
| S | A127093 |     |       |         |  1        |  1 2        |
| O | A127093 |     |       |         |  1        |  1 2        |
| R |---------|-----|-------|---------|-----------|-------------|
| S | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A127093 |  1  |  1 2  |  1 0 3  |  1 2 0 4  |  1 0 0 0 5  |
| C | A127093 |     |  1    |  1 2    |  1 0 3    |  1 2 0 4    |
| O |    -    |     |       |  2      |  2 4      |  2 0 6      |
| N |    -    |     |       |         |  3        |  3 6        |
| D |    -    |     |       |         |           |  5          |
|---|---------|-----|-------|---------|-----------|-------------|
.
This lower zone of the table is a condensed version of the "divisors" zone.

Paolo Xausa, Table of n, a(n) for n = 1..11480 (rows 1..40 of the triangle, flattened)

Row sums give A066186.
Nonzero terms give A340056.
Cf. A000070, A000041, A002260, A026792, A027750, A058399, A066633, A127093, A135010, A138121, A138785, A176206, A181187, A182703, A207031, A207383, A211992, A221529, A221530, A221531, A221649, A221650, A237593, A245095, A302246, A302247, A336811, A336812, A337209, A338156, A339106, A339258, A339278, A339304, A340031, A340032, A340035, A340061.

A127093row[n_]:=Table[Boole[Divisible[n,k]]k,{k,n}];
A340011row[n_]:=Flatten[Table[A127093row[n-m+1]PartitionsP[m-1],{m,n}]];
Array[A340011row,10] (* Paolo Xausa, Sep 28 2023 *)

A340032 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of A000041(n-m) copies of the row m of triangle A127093, with 1 <= m <= n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 1, 0, 3, 1, 1, 1, 1, 2, 1, 2, 1, 0, 3, 1, 2, 0, 4, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 0, 3, 1, 0, 3, 1, 2, 0, 4, 1, 0, 0, 0, 5, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 0, 3, 1, 0, 3, 1, 0, 3, 1, 2, 0, 4, 1, 2, 0, 4, 1, 0, 0, 0, 5, 1, 2, 3, 0, 0, 6

Offset: 1

Omar E. Pol, Dec 26 2020

For further information about the correspondence divisor/part see A338156.

			Triangle begins:
  1;
  1, 1, 2;
  1, 1, 1, 2, 1, 0, 3;
  1, 1, 1, 1, 2, 1, 2, 1, 0, 3, 1, 2, 0, 4;
  1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 0, 3, 1, 0, 3, 1, 2, 0, 4, 1, 0, 0, 0, 5;
  ...
Written as an irregular tetrahedron the first five slices are:
  1;
  --
  1,
  1, 2;
  -----
  1,
  1,
  1, 2,
  1, 0, 3;
  --------
  1,
  1,
  1,
  1, 2,
  1, 2,
  1, 0, 3,
  1, 2, 0, 4;
  -----------
  1,
  1,
  1,
  1,
  1,
  1, 2,
  1, 2,
  1, 2,
  1, 0, 3,
  1, 0, 3,
  1, 2, 0, 4,
  1, 0, 0, 0, 5;
  --------------
  ...
The slices of the tetrahedron appear in the upper zone of the following table (formed by three zones) which shows the correspondence between divisors and parts (n = 1..5):
.
|---|---------|-----|-------|---------|-----------|-------------|
| n |         |  1  |   2   |    3    |     4     |      5      |
|---|---------|-----|-------|---------|-----------|-------------|
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
| D | A127093 |     |       |         |           |  1          |
| I |---------|-----|-------|---------|-----------|-------------|
| V | A127093 |     |       |         |  1        |  1 2        |
| I | A127093 |     |       |         |  1        |  1 2        |
| S | A127093 |     |       |         |  1        |  1 2        |
| O |---------|-----|-------|---------|-----------|-------------|
| R | A127093 |     |       |  1      |  1 2      |  1 0 3      |
| S | A127093 |     |       |  1      |  1 2      |  1 0 3      |
|   |---------|-----|-------|---------|-----------|-------------|
|   | A127093 |     |  1    |  1 2    |  1 0 3    |  1 2 0 4    |
|   |---------|-----|-------|---------|-----------|-------------|
|   | A127093 |  1  |  1 2  |  1 0 3  |  1 2 0 4  |  1 0 0 0 5  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A138785 |  1  |  2 2  |  4 2 3  |  7 6 3 4  | 12 8 6 4 5  |
|   |         |  =  |  = =  |  = = =  |  = = = =  |  = = = = =  |
| L | A002260 |  1  |  1 2  |  1 2 3  |  1 2 3 4  |  1 2 3 4 5  |
| I |         |  *  |  * *  |  * * *  |  * * * *  |  * * * * *  |
| N | A066633 |  1  |  2 1  |  4 1 1  |  7 3 1 1  | 12 4 2 1 1  |
| K |         |  |  |  |\|  |  |\|\|  |  |\|\|\|  |  |\|\|\|\|  |
|   | A181187 |  1  |  3 1  |  6 2 1  | 12 5 2 1  | 20 8 4 2 1  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
| P |         |  1  |  1 1  |  1 1 1  |  1 1 1 1  |  1 1 1 1 1  |
| A |         |     |  2    |  2 1    |  2 1 1    |  2 1 1 1    |
| R |         |     |       |  3      |  3 1      |  3 1 1      |
| T |         |     |       |         |  2 2      |  2 2 1      |
| I |         |     |       |         |  4        |  4 1        |
| T |         |     |       |         |           |  3 2        |
| I |         |     |       |         |           |  5          |
| O |         |     |       |         |           |             |
| N |         |     |       |         |           |             |
| S |         |     |       |         |           |             |
|---|---------|-----|-------|---------|-----------|-------------|
.
The table is essentially the same table of A340035 but here, in the upper zone, every row is A127093 instead of A027750.
Also the above table is the table of A340031 upside down.

Paolo Xausa, Table of n, a(n) for n = 1..11552 (rows 1..17 of the triangle, flattened)

Row sums give A066186.
Nonzero terms gives A340035.
Cf. A000070, A000041, A002260, A026792, A027750, A058399, A066633, A127093, A135010, A138121, A138785, A176206, A181187, A182703, A207031, A207383, A211992, A221529, A221530, A221531, A245095, A221649, A221650, A237593, A302246, A302247, A336811, A337209, A338156, A339106, A339258, A339278, A339304, A340031, A340061.

A127093row[n_]:=Table[Boole[Divisible[n,k]]k,{k,n}];
A340032row[n_]:=Flatten[Table[ConstantArray[A127093row[m],PartitionsP[n-m]],{m,n}]];
Array[A340032row,7] (* Paolo Xausa, Sep 28 2023 *)

cp's OEIS Frontend

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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A340031 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of A000041(m-1) copies of the j-th row of triangle A127093, where j = n - m + 1 and 1 <= m <= n.

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A182713 Number of 3's in the last section of the set of partitions of n.

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A207034 Sum of all parts minus the number of parts of the n-th partition in the list of colexicographically ordered partitions of j, if 1<=n<=A000041(j).

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A210979 Total area of the shadows of the three views of the version "Tree" of the shell model of partitions with n shells.

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A340011 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of the j-th row of triangle A127093 but with every term multiplied by A000041(m-1), where j = n - m + 1 and 1 <= m <= n.

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A340032 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of A000041(n-m) copies of the row m of triangle A127093, with 1 <= m <= n.