A141285 -id:A141285 - cp's OEIS frontend

A210990 Total area of the shadows of the three views of the shell model of partitions with n regions.

0, 3, 10, 21, 26, 44, 51, 75, 80, 92, 99, 136, 143, 157, 166, 213, 218, 230, 237, 260, 271, 280, 348, 355, 369, 378, 403, 410, 427, 438, 526, 531, 543, 550, 573, 584, 593, 631, 640, 659, 672, 683, 804, 811, 825, 834, 859, 866, 883, 894, 938, 949, 958

Offset: 0

Omar E. Pol, Apr 23 2012

nonn

Each part is represented by a cuboid of sides 1 X 1 X k where k is the size of the part. For the definition of "regions of n" see A206437.

			For n = 11 the three views of the shell model of partitions with 11 regions look like this:
.
.   A182181(11) = 35            A182244(11) = 66
.
.   6                             * * * * * 6
.   3 3                      P    * * 3 * * 3
.   2   4                    a    * * * 4 * 2
.   2   2 2                  r    * 2 * 2 * 2
.   1       5                t    * * * * 5 1
.   1       2 3              i    * * 3 * 2 1
.   1       1   4            t    * * * 4 1 1
.   1       1   2 2          i    * 2 * 2 1 1
.   1       1   1   3        o    * * 3 1 1 1
.   1       1   1   1 2      n    * 2 1 1 1 1
.   1       1   1   1 1 1    s    1 1 1 1 1 1
. <------- Regions ------         ------------> N
.                            L
.                            a    1
.                            r    * 2
.                            g    * * 3
.                            e    * 2
.                            s    * * * 4
.                            t    * * 3
.                                 * * * * 5
.                            p    * 2
.                            a    * * * 4
.                            r    * * 3
.                            t    * * * * * 6
.                            s
.                               A182727(11) = 35
.
The areas of the shadows of the three views are A182244(11) = 66, A182181(11) = 35 and A182727(11) = 35, therefore the total area of the three shadows is 66+35+35 = 136, so a(11) = 136.

Other versions: A207380, A210970, A210979, A210980, A210991.

Cf. A135010, A138121, A141285, A194446, A182181, A182727, A186114, A206437.

a(n) = A182244(n) + A182727(n) + A182181(n), n >= 1.

a(A000041(n)) = 2*A006128(n) + A066186(n).

A228354 Indices (k) of partitions in the list of compositions of j in colexicographic order, if 1<=k<=2^(j-1), j>=1.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 22, 24, 28, 32, 44, 48, 56, 64, 86, 88, 92, 96, 112, 120, 128, 172, 176, 184, 192, 220, 224, 240, 256, 342, 344, 348, 352, 368, 376, 384, 440, 448, 480, 496, 512, 684, 688, 696, 704, 732, 736, 752, 768, 880, 888, 896, 960, 992, 1024

Offset: 1

Omar E. Pol, Aug 20 2013

Also where records occur in A228720.
Also triangle read by rows in which row j lists the indices of the partitions of j into parts greater than the smallest part of the partitions of j-1, in the list of compositions of j in colexicographic order. See A228525 and A211992.
The total number of terms in the first j rows of triangle is A000041(j), j >= 1.
Row j has length A187219(j).
Right border gives A000079.

			For j = 5 consider the list of compositions of 5 in colexicographic order (see A228525). The indices of the partitions are 1, 2, 4, 6, 8, 12, 16 which are the first A000041(5) terms of this sequence, see below:
---------------------------------------------------------
.   Compositions                     Partitions
k      of 5                             of 5      n  a(n)
---------------------------------------------------------
1    1+1+1+1+1  * ............... *  1+1+1+1+1    1    1
2    2+1+1+1    * ............... *  2+1+1+1      2    2
3    1+2+1+1          ........... *  3+1+1        3    4
4    3+1+1      * .../ .......... *  2+2+1        4    6
5    1+1+2+1          / ......... *  4+1          5    8
6    2+2+1      * .../ /   ...... *  3+2          6   12
7    1+3+1            /   /   ... *  5            7   16
8    4+1        * .../   /   /
9    1+1+1+2            /   /
10   2+1+2             /   /
11   1+2+2            /   /
12   3+2        * .../   /
13   1+1+3              /
14   2+3               /
15   1+4              /
16   5          * .../
.
Written as an irregular triangle the sequence begins:
1;
2;
4;
6,8;
12,16;
22,24,28,32;
44,48,56,64;
86,88,92,96,112,120,128;
172,176,184,192,220,224,240,256;
342,344,348,352,368,376,384,440,448,480,496,512;
684,688,696,704,732,736,752,768,880,888,896,960,992,1024;
...

Cf. A000041, A187219, A211992, A228354, A228525, A228720.

a(n) = 1 + A194602(n-1).
A001511(a(n)) = A141285(n).
A000120(a(n)-1) = A207034(n).

A194714 Sum of all odd-indexed parts minus the sum of all even-indexed parts of all partitions of n, with the parts written in nondecreasing order.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 14, 18, 26, 32, 48, 57, 82, 102, 138, 169, 230, 278, 370, 450, 584, 709, 914, 1102, 1400, 1692, 2124, 2555, 3186, 3818, 4720, 5649, 6926, 8269, 10078, 11989, 14526, 17249, 20782, 24603, 29508, 34843, 41600, 49008, 58258, 68468, 81098

Offset: 1

Omar E. Pol, Feb 12 2012

nonn

It appears that A066897 is also another version of this sequence but with the parts written in nonincreasing order.

			a(6) = 37 - 29 = 8 because the partitions of 6 written in nondecreasing order are
.
.   6                        =  6
.   3  - 3                   =  0
.   2  - 4                   = -2
.   2  - 2  + 2              =  2
.   1  - 5                   = -4
.   1  - 2  + 3              =  2
.   1  - 1  + 4              =  4
.   1  - 1  + 2 - 2          =  0
.   1  - 1  + 1 - 3          = -2
.   1  - 1  + 1 - 1 + 2      =  2
.   1  - 1  + 1 - 1 + 1 - 1  =  0
----------------------------------
.  20 - 21 + 14 - 7 + 3 - 1  =  8

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

Cf. A066897, A135010, A138121, A141285, A194853.

More terms from Alois P. Heinz, Feb 12 2012

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).

A220482 Triangle read by rows: T(j,k) in which row j lists the parts in nondecreasing order of the j-th region of the set of partitions of n, with 1<=j<=A000041(n).

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 4, 3, 1, 1, 1, 1, 1, 2, 5, 2, 2, 4, 3, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 6, 3, 2, 5, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 7, 2, 2, 4, 3, 2, 2, 3, 6, 5, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 8

Offset: 1

Omar E. Pol, Jan 27 2013

For the definition of "region" of the set of partitions of n see A206437.

			First 15 rows of the irregular triangle are
1;
1, 2;
1, 1, 3;
2;
1, 1, 1, 2, 4;
3;
1, 1, 1, 1, 1, 2, 5;
2;
2, 4;
3;
1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 6;
3;
2, 5;
4;
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 7;

Positive terms of A186114. Mirror of A206437.
Row j has length A194446(j). Row sums give A186412.
Cf. A135010, A141285, A193870.

A228350 Triangle read by rows: T(j,k) is the k-th part in nonincreasing order of the j-th region of the set of compositions (ordered partitions) of n in colexicographic order, if 1<=j<=2^(n-1) and 1<=k<=A006519(j).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 20 2013

Triangle read by rows in which row n lists the A006519(n) elements of the row A001511(n) of triangle A065120, n >= 1.

The equivalent sequence for integer partitions is A206437.

			---------------------------------------------------------
.              Diagram                Triangle
Compositions     of            of compositions (rows)
.   of 5       regions          and regions (columns)
----------------------------------------------------------
.             _ _ _ _ _
.         5  |_        |                                5
.       1+4  |_|_      |                              1 4
.       2+3  |_  |     |                            2   3
.     1+1+3  |_|_|_    |                          1 1   3
.       3+2  |_    |   |                        3       2
.     1+2+2  |_|_  |   |                      1 2       2
.     2+1+2  |_  | |   |                    2   1       2
.   1+1+1+2  |_|_|_|_  |                  1 1   1       2
.       4+1  |_      | |                4               1
.     1+3+1  |_|_    | |              1 3               1
.     2+2+1  |_  |   | |            2   2               1
.   1+1+2+1  |_|_|_  | |          1 1   2               1
.     3+1+1  |_    | | |        3       1               1
.   1+2+1+1  |_|_  | | |      1 2       1               1
.   2+1+1+1  |_  | | | |    2   1       1               1
. 1+1+1+1+1  |_|_|_|_|_|  1 1   1       1               1
.
Also the structure could be represented by an isosceles triangle in which the n-th diagonal gives the n-th region. For the composition of 4 see below:
.             _ _ _ _
.         4  |_      |                  4
.       1+3  |_|_    |                1   3
.       2+2  |_  |   |              2       2
.     1+1+2  |_|_|_  |            1   1       2
.       3+1  |_    | |          3               1
.     1+2+1  |_|_  | |        1   2               1
.     2+1+1  |_  | | |      2       1               1
.   1+1+1+1  |_|_|_|_|    1   1       1               1
.
Illustration of the four sections of the set of compositions of 4:
.                                      _ _ _ _
.                                     |_      |     4
.                                     |_|_    |   1+3
.                                     |_  |   |   2+2
.                       _ _ _         |_|_|_  | 1+1+2
.                      |_    |   3          | |     1
.             _ _      |_|_  | 1+2          | |     1
.     _      |_  | 2       | |   1          | |     1
.    |_| 1     |_| 1       |_|   1          |_|     1
.
.
Illustration of initial terms. The parts of the eight regions of the set of compositions of 4:
--------------------------------------------------------
\j:  1      2    3        4     5      6    7          8
k
--------------------------------------------------------
.  _    _ _    _    _ _ _     _    _ _    _    _ _ _ _
1 |_|1 |_  |2 |_|1 |_    |3  |_|1 |_  |2 |_|1 |_      |4
2        |_|1        |_  |2         |_|1        |_    |3
3                      | |1                       |   |2
4                      |_|1                       |_  |2
5                                                   | |1
6                                                   | |1
7                                                   | |1
8                                                   |_|1
.
Triangle begins:
1;
2,1;
1;
3,2,1,1;
1;
2,1;
1;
4,3,2,2,1,1,1,1;
1;
2,1;
1;
3,2,1,1;
1;
2,1;
1;
5,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1;
...
.
Also triangle read by rows T(n,m) in which row n lists the parts of the n-th section of the set of compositions of the integers >= n, ordered by regions. Row lengths give A045623. Row sums give A001792 (see below):
[1];
[2,1];
[1],[3,2,1,1];
[1],[2,1],[1],[4,3,2,2,1,1,1,1];
[1],[2,1],[1],[3,2,1,1],[1],[2,1],[1],[5,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1];

Main triangle: column 1 is A001511. Row j has length A006519(j). Row sums give A038712.

Cf. A001787, A001792, A011782, A029837, A045623, A065120, A070939, A135010, A141285, A187816, A187818, A193870, A206437, A228347, A228348, A228349, A228351, A228366, A228367, A228370, A228371, A228525, A228526.

T(j,k) = A065120(A001511(j)),k) = A001511(j) - A029837(k), 1<=k<=A006519(j), j>=1.

A210980 Total area of the shadows of the three views of the shell model of partitions, version "Tree", with n shells.

Original entry on oeis.org

0, 3, 10, 21, 42, 69, 123, 189, 304, 458, 693, 998, 1474, 2067, 2927, 4056, 5613, 7595, 10335, 13782, 18411, 24276, 31944, 41583, 54152, 69762, 89758, 114668, 146181, 185083, 234051, 294126, 368992, 460669, 573906, 711865, 881506, 1087023, 1338043

Offset: 0

Omar E. Pol, Apr 21 2012

nonn

Each part is represented by a cuboid 1 X 1 X L where L is the size of the part.

			For n = 7 the shadows of the three views of the shell model of partitions version "tree" with seven shells looks like this:
.                                        |  Partitions
.    A194805(7) = 25    A066186(7) = 105 |  of 7
.                                        |
.                   1    * * * * * * 1   |  7
.                 2      * * * 1 * * 2   |  4+3
.               2        * * * * 1 * 2   |  5+2
.             3          * * 1 * 2 * 3   |  3+2+2
.   1       2            * * * * * 1 2   |  6+1
.     2     3            * * 1 * * 2 3   |  3+3+1
.       2   3            * * * 1 * 2 3   |  4+2+1
.         3 4            * 1 * 2 * 3 4   |  2+2+2+1
.           3   1        * * * * 1 2 3   |  5+1+1
.           4 2          * * 1 * 2 3 4   |  3+2+1+1
.       1   4            * * * 1 2 3 4   |  4+1+1+1
.         2 5            * 1 * 2 3 4 5   |  2+2+1+1+1
.           5 1          * * 1 2 3 4 5   |  3+1+1+1+1
.         1 6            * 1 2 3 4 5 6   |  2+1+1+1+1+1
.           7            1 2 3 4 5 6 7   |  1+1+1+1+1+1+1
.   ----------------------------------   |
.                                        |
.   * * * * 1 * * * *                    |
.   * * * 1 2 * * * *                    |
.   * 1 * * 2 1 * * *                    |
.   * * 1 2 2 * * 1 *                    |
.   * * * * 2 2 1 * *                    |
.   1 2 2 3 2 * * * *                    |
.           2 3 2 2 1                    |
.                                        |
.    A194804(7) = 59                     |
.
Note that, as a variant, in this case each part is labeled with its position in the partition.
The areas of the shadows of the three views are A066186(7) = 105, A194804(7) = 59 and A194805(7) = 25, therefore the total area of the three shadows is 105+59+25 = 189, so a(7) = 189.

Other versions: A207380, A210970, A210979, A210990, A210991.

Cf. A000041, A066186, A135010, A138121, A141285, A182703, A194804, A194805, A206437.

a(n) = A066186(n) + A194804(n) + A194805(n), n >= 1.

A210991 Total area of the shadows of the three views of the shell model of partitions with n regions.

Original entry on oeis.org

0, 3, 9, 18, 21, 35, 39, 58, 61, 67, 71, 99, 103, 110, 115, 152, 155, 161, 165, 175, 181, 186, 238, 242, 249, 254, 265, 269, 277, 283, 352, 355, 361, 365, 375, 381, 386, 401, 406, 415, 422, 428, 522, 526, 533, 538, 549, 553, 561, 567, 584, 590, 595, 606

Offset: 0

Omar E. Pol, Apr 30 2012

nonn

It appears that if n is a partition number A000041 then the rotated structure with n regions shows each row as a partition of k such that A000041(k) = n (see example).

For the definition of "regions of n" see A206437.

			For n = 11 the three views of the shell model of partitions with 11 regions look like this:
.
.     A182181(11) = 35           A210692(11) = 29
.
.   1                                       1
.   1                                       1
.   1                                       1
.   1                                       1
.   1       1                             1 1
.   1       1                             1 1
.   1       1   1                       1 1 1
.   2       1   1                       1 1 2
.   2       1   1   1                 1 1 1 2
.   3   2   2   2   1 1             1 1 2 2 3
.   6 3 4 2 5 3 4 2 3 2 1         1 2 3 4 5 6
. <------- Regions ------         ------------> N
.                            L
.                            a    1
.                            r    * 2
.                            g    * * 3
.                            e    * 2
.                            s    * * * 4
.                            t    * * 3
.                                 * * * * 5
.                            p    * 2
.                            a    * * * 4
.                            r    * * 3
.                            t    * * * * * 6
.                            s
.
.                                A182727(11) = 35
.
The areas of the shadows of the three views are A182181(11) = 35, A182727(11) = 35 and A210692(11) = 29, therefore the total area of the three shadows is 35+35+29 = 99, so a(11) = 99.
Since n = 11 is a partition number A000041 we can see that the rotated structure with 11 regions shows each row as a partition of 6 because A000041(6) = 11. See below:
.
.                      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
.

Other versions: A207380, A210970, A210979, A210980, A210990.

Cf. A000041, A026905, A135010, A138121, A141285, A182703, A194446, A182181, A182727, A186114, A206437, A210692.

a(n) = A182181(n) + A182727(n) + A210692(n).

a(A000041(n)) = 2*A006128(n) + A026905(n).

A278355 a(n) = sum of the perimeters of the Ferrers boards of the partitions of n. Also, sum of the perimeters of the diagrams of the regions of the set of partitions of n.

Original entry on oeis.org

0, 4, 12, 24, 48, 80, 140, 216, 344, 512, 768, 1100, 1596, 2224, 3120, 4272, 5852, 7860, 10576, 13992, 18520, 24208, 31596, 40824, 52696, 67404, 86088, 109176, 138180, 173812, 218252, 272540, 339708, 421464, 521848, 643504, 792056, 971248, 1188804, 1450348, 1766184, 2144416, 2599164, 3141748, 3791248, 4563780

Offset: 0

Omar E. Pol, Nov 19 2016

nonn

a(n) is also 4 times the total number of parts in all partitions of n.
Hence a(n) is also 4 times the sum of largest parts of all partitions of n.
Hence a(n) is also twice the total number of parts in all partitions of n plus twice the sum of largest parts of all partitions of n.
a(n) is also the sum of the perimeters of the first n polygons constructed with the Dyck path (and its mirror) that arises from the minimalist diagram of the regions of the set of partitions of n. The n-th odd-indexed segment of the diagram has A141285(n) up-steps and the n-th even-indexed segment has A194446(n) down-steps. The k-th polygon of the diagram is associated to the k-th section of the set of partitions of n, with 1<=k<=n. See the bottom of Example section. For the definition of "section" see A135010. For the definition of "region" see A206437.

			For n = 5 consider the partitions of 5 in colexicographic order (as shown in the 5th row of the triangle A211992) and its associated diagram of regions as shown below:
.                                Regions            Minimalist
.         Partitions of 5        diagram             version
.                               _ _ _ _ _
.         1, 1, 1, 1, 1        |_| | | | |          _| | | | |
.         2, 1, 1, 1           |_ _| | | |          _ _| | | |
.         3, 1, 1              |_ _ _| | |          _ _ _| | |
.         2, 2, 1              |_ _|   | |          _ _|   | |
.         4, 1                 |_ _ _ _| |          _ _ _ _| |
.         3, 2                 |_ _ _|   |          _ _|     |
.         5                    |_ _ _ _ _|          _ _ _ _ _|
.
Then consider the following table which contains the Ferrers boards of the partitions of 5 and the diagram of every region of the set of partitions of 5:
-------------------------------------------------------------------------
| Partitions  |             |       |   Regions   |             |       |
|     of 5    |   Ferrers   | Peri- |     of 5    |   Region    | Peri- |
|(See A211992)|    board    | meter |(see A220482)|   diagram   | meter |
-------------------------------------------------------------------------
|                  _                |                 _                 |
|      1          |_|               |       1        |_|            4   |
|      1          |_|               |                   _               |
|      1          |_|               |       1         _|_|              |
|      1          |_|               |       2        |_|_|          8   |
|      1          |_|          12   |                     _             |
|                  _ _              |       1            |_|            |
|      2          |_|_|             |       1         _ _|_|            |
|      1          |_|               |       3        |_|_|_|       12   |
|      1          |_|               |                 _ _               |
|      1          |_|          12   |       2        |_|_|          6   |
|                  _ _ _            |                       _           |
|      3          |_|_|_|           |       1              |_|          |
|      1          |_|               |       1              |_|          |
|      1          |_|          12   |       1             _|_|          |
|                  _ _              |       2         _ _|_|_|          |
|      2          |_|_|             |       4        |_|_|_|_|     18   |
|      2          |_|_|             |                 _ _ _             |
|      1          |_|          10   |       3        |_|_|_|        8   |
|                  _ _ _ _          |                         _         |
|      4          |_|_|_|_|         |       1                |_|        |
|      1          |_|          12   |       1                |_|        |
|                  _ _ _            |       1                |_|        |
|      3          |_|_|_|           |       1                |_|        |
|      2          |_|_|        10   |       1               _|_|        |
|                  _ _ _ _ _        |       2         _ _ _|_|_|        |
|      6          |_|_|_|_|_|  12   |       5        |_|_|_|_|_|   24   |
|                                   |                                   |
-------------------------------------------------------------------------
|   Sum of perimeters:         80         <-- equals -->           80   |
-------------------------------------------------------------------------
The sum of the perimeters of the Ferrers boards is 12 + 12 + 12 + 10 + 12 + 10 + 12 = 80, so a(5) = 80.
On the other hand, the sum of the perimeters of the diagrams of regions is 4 + 8 + 12 + 6 + 18 + 8 + 24 = 80, equaling the sum of the perimeters of the Ferrers boards.
.
Illustration of first six polygons of an infinite diagram constructed with the boundary segments of the minimalist diagram of regions and its mirror (note that the diagram looks like reflections on a mountain lake):
11............................................................
.                                                            /\
.                                                           /  \
.                                                          /    \
7...................................                      /      \
.                                  /\                    /        \
5.....................            /  \                /\/          \
.                    /\          /    \          /\  /              \
3...........        /  \        /      \        /  \/                \
2.......   /\      /    \    /\/        \      /                      \
1...  /\  /  \  /\/      \  /            \  /\/                        \
0  /\/  \/    \/          \/              \/                            \
.  \/\  /\    /\          /\              /\                            /
.     \/  \  /  \/\      /  \            /  \/\                        /
.          \/      \    /    \/\        /      \                      /
.                   \  /        \      /        \  /\                /
.                    \/          \    /          \/  \              /
.                                 \  /                \/\          /
.                                  \/                    \        /
.                                                         \      /
.                                                          \    /
.                                                           \  /
.                                                            \/
n:
. 0 1  2   3          4             5                         6
Perimeter of the n-th polygon:
. 0 4  8  12         24            32                        60
a(n) is the sum of the perimeters of the first n polygons:
. 0 4 12  24         48            80                       140
.
For n = 5, the sum of the perimeters of the first five polygons is 4 + 8 + 12 + 24 + 32 = 80, so a(5) = 80.
For n = 6, the sum of the perimeters of the first six polygons is 4 + 8 + 12 + 24 + 32 + 60 = 140, so a(6) = 140.
For another version of the above diagram see A228109.

Cf. A000041, A006128, A135010, A138137, A139582, A141285, A194446, A211992, A220482, A225600, A211978, A233968, A244968.

a(n) = 4*A006128(n) = 2*A211978(n).
a(n) = 2*A225600(2*A000041(n)) = 2*A225600(A139582(n)), n >= 1.
a(n) = 2*((Sum_{m=1..p(n)} A194446(m)) + (Sum_{m=1..p(n)} A141285(m))) = 4*Sum_{m=1..p(n)} A194446(m) = 4*Sum_{m=1..p(n)} A141285(m), where p(n) = A000041(n), n >= 1.

A299474 a(n) = 4*p(n), where p(n) is the number of partitions of n.

Original entry on oeis.org

4, 4, 8, 12, 20, 28, 44, 60, 88, 120, 168, 224, 308, 404, 540, 704, 924, 1188, 1540, 1960, 2508, 3168, 4008, 5020, 6300, 7832, 9744, 12040, 14872, 18260, 22416, 27368, 33396, 40572, 49240, 59532, 71908, 86548, 104060, 124740, 149352, 178332, 212696, 253044, 300700, 356536, 422232, 499016, 589092, 694100, 816904

Offset: 0

Omar E. Pol, Feb 10 2018

nonn

For n >= 1, a(n) is also the number of edges in the diagram of partitions of n, in which A299475(n) is the number of vertices and A000041(n) is the number of regions (see example and Euler's formula).

			Construction of a modular table of partitions in which a(n) is the number of edges of the diagram after n-th stage (n = 1..6):
--------------------------------------------------------------------------------
n ........:   1     2       3         4           5             6     (stage)
a(n)......:   4     8      12        20          28            44     (edges)
A299475(n):   4     7      10        16          22            34     (vertices)
A000041(n):   1     2       3         5           7            11     (regions)
--------------------------------------------------------------------------------
r     p(n)
--------------------------------------------------------------------------------
.             _    _ _    _ _ _    _ _ _ _    _ _ _ _ _    _ _ _ _ _ _
1 .... 1 ....|_|  |_| |  |_| | |  |_| | | |  |_| | | | |  |_| | | | | |
2 .... 2 .........|_ _|  |_ _| |  |_ _| | |  |_ _| | | |  |_ _| | | | |
3 .... 3 ................|_ _ _|  |_ _ _| |  |_ _ _| | |  |_ _ _| | | |
4                                 |_ _|   |  |_ _|   | |  |_ _|   | | |
5 .... 5 .........................|_ _ _ _|  |_ _ _ _| |  |_ _ _ _| | |
6                                            |_ _ _|   |  |_ _ _|   | |
7 .... 7 ....................................|_ _ _ _ _|  |_ _ _ _ _| |
8                                                         |_ _|   |   |
9                                                         |_ _ _ _|   |
10                                                        |_ _ _|     |
11 .. 11 .................................................|_ _ _ _ _ _|
.
Apart from the axis x, the r-th horizontal line segment has length A141285(r), equaling the largest part of the r-th region of the diagram.
Apart from the axis y, the r-th vertical line segment has length A194446(r), equaling the number of parts in the r-th region of the diagram.
The total number of parts equals the sum of largest parts.
Note that every diagram contains all previous diagrams.
An infinite diagram is a table of all partitions of all positive integers.

Shawn A. Broyles, Table of n, a(n) for n = 0..1000

k times partition numbers: A000041 (k=1), A139582 (k=2), A299473 (k=3), this sequence (k=4).

Cf. A135010, A141285, A182181, A186114, A193870, A194446, A194447, A206437, A207779, A220482, A220517, A273140, A278355, A278602, A299475.

List([0..50],n->4*NrPartitions(n)); # Muniru A Asiru, Jul 10 2018

with(combinat): seq(4*numbpart(n),n=0..50); # Muniru A Asiru, Jul 10 2018

4*PartitionsP[Range[0,50]] (* Harvey P. Dale, Dec 05 2023 *)

a(n) = 4*numbpart(n); \\ Michel Marcus, Jul 15 2018

from sympy.ntheory import npartitions
def a(n): return 4*npartitions(n)
print([a(n) for n in range(51)]) # Michael S. Branicky, Apr 04 2021

a(n) = 4*A000041(n) = 2*A139582(n).
a(n) = A000041(n) + A299475(n) - 1, n >= 1 (Euler's formula).
a(n) = A000041(n) + A299473(n). - Omar E. Pol, Aug 11 2018

cp's OEIS Frontend

A210990 Total area of the shadows of the three views of the shell model of partitions with n regions.

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A228354 Indices (k) of partitions in the list of compositions of j in colexicographic order, if 1<=k<=2^(j-1), j>=1.

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A194714 Sum of all odd-indexed parts minus the sum of all even-indexed parts of all partitions of n, with the parts written in nondecreasing order.

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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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A220482 Triangle read by rows: T(j,k) in which row j lists the parts in nondecreasing order of the j-th region of the set of partitions of n, with 1<=j<=A000041(n).

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A228350 Triangle read by rows: T(j,k) is the k-th part in nonincreasing order of the j-th region of the set of compositions (ordered partitions) of n in colexicographic order, if 1<=j<=2^(n-1) and 1<=k<=A006519(j).

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A210980 Total area of the shadows of the three views of the shell model of partitions, version "Tree", with n shells.

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A210991 Total area of the shadows of the three views of the shell model of partitions with n regions.

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A278355 a(n) = sum of the perimeters of the Ferrers boards of the partitions of n. Also, sum of the perimeters of the diagrams of the regions of the set of partitions of n.

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A299474 a(n) = 4*p(n), where p(n) is the number of partitions of n.

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