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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Views

Author

Omar E. Pol, Feb 10 2018

Keywords

Comments

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).

Examples

			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.

Links

Crossrefs

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.

Programs

  • GAP
    List([0..50],n->4*NrPartitions(n)); # Muniru A Asiru, Jul 10 2018
  • Maple
    with(combinat): seq(4*numbpart(n),n=0..50); # Muniru A Asiru, Jul 10 2018
  • Mathematica
    4*PartitionsP[Range[0,50]] (* Harvey P. Dale, Dec 05 2023 *)
  • PARI
    a(n) = 4*numbpart(n); \\ Michel Marcus, Jul 15 2018
  • Python
    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

Formula

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

A299475 a(n) is the number of vertices in the diagram of partitions of n (see example).

Original entry on oeis.org

1, 4, 7, 10, 16, 22, 34, 46, 67, 91, 127, 169, 232, 304, 406, 529, 694, 892, 1156, 1471, 1882, 2377, 3007, 3766, 4726, 5875, 7309, 9031, 11155, 13696, 16813, 20527, 25048, 30430, 36931, 44650, 53932, 64912, 78046, 93556, 112015, 133750, 159523, 189784, 225526, 267403, 316675, 374263, 441820, 520576, 612679
Offset: 0

Views

Author

Omar E. Pol, Feb 11 2018

Keywords

Comments

For n >= 1, A299474(n) is the number of edges and A000041(n) is the number of regions in the mentioned diagram (see example and Euler's formula).

Examples

			Construction of a modular table of partitions in which a(n) is the number of vertices of the diagram after n-th stage (n = 1..6):
--------------------------------------------------------------------------------
n ........:   1     2       3         4           5             6     (stage)
a(n)......:   4     7      10        16          22            34     (vertices)
A299474(n):   4     8      12        20          28            44     (edges)
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.

Links

Crossrefs

Cf. A000041, A135010, A139582, A141285, A182181, A186114, A193870, A194446, A194447, A206437, A207779, A220482, A220517, A273140, A278355, A278602, A299474.

Programs

  • PARI
    a(n) = if (n==0, 1, 1+3*numbpart(n)); \\ Michel Marcus, Jul 15 2018

Formula

a(0) = 1; a(n) = 1 + 3*A000041(n), n >= 1.
a(n) = A299474(n) - A000041(n) + 1, n >= 1 (Euler's formula).

A278602 Sum of the perimeters of all regions of the n-th section of a modular table of partitions.

Original entry on oeis.org

0, 4, 8, 12, 24, 32, 60, 76, 128, 168, 256, 332, 496, 628, 896, 1152, 1580, 2008, 2716, 3416, 4528, 5688, 7388, 9228, 11872, 14708, 18684, 23088, 29004, 35632, 44440, 54288, 67168, 81756, 100384, 121656, 148552, 179192, 217556, 261544, 315836, 378232, 454748, 542584, 649500, 772532, 920912
Offset: 0

Views

Author

Omar E. Pol, Nov 23 2016

Keywords

Comments

Consider an infinite dissection of the fourth quadrant of the square grid in which, apart from the axes x and y, the k-th horizontal line segment has length A141285(k) and the k-th vertical line segment has length A194446(k). Both line segments shares the point (A141285(k),k). For n>=1, the table contains A000041(n) regions which are distributed in n sections. Note that in the infinite table there are no partitions because every row contains an infinite number of parts. On the other hand, taking only the first n sections from the table we have a representation of the partitions of n. For an illustration see the example. For the definition of "region" see A206437. For the definition of "section" see A135010. For a visualization of the corner of size n X n of the table see A273140.
a(n) is also the sum of the perimeters of the Ferrers boards of the partitions of n, minus the sum of the perimeters of the Ferrers boards of the partitions of n-1, with n >= 1. For more information see A278355.

Examples

			For n = 1..6, consider the modular table of partitions for the first six positive integers as shown below in the fourth quadrant of the square grid (see Figure 1):
|--------------|-----------------------------------------------------|
| Modular table|                      Sections                       |
| of partitions|-----------------------------------------------------|
|  for n=1..6  | 1     2       3         4           5             6 |
1--------------|-----------------------------------------------------|
.  _ _ _ _ _ _   _     _       _         _           _             _
. |_| | | | | | |_|  _| |     | |       | |         | |           | |
. |_ _| | | | |     |_ _|  _ _| |       | |         | |           | |
. |_ _ _| | | |           |_ _ _|  _ _ _| |         | |           | |
. |_ _|   | | |                   |_ _|   |         | |           | |
. |_ _ _ _| | |                   |_ _ _ _|  _ _ _ _| |           | |
. |_ _ _|   | |                             |_ _ _|   |           | |
. |_ _ _ _ _| |                             |_ _ _ _ _|  _ _ _ _ _| |
. |_ _|   |   |                                         |_ _|   |   |
. |_ _ _ _|   |                                         |_ _ _ _|   |
. |_ _ _|     |                                         |_ _ _|     |
. |_ _ _ _ _ _|                                         |_ _ _ _ _ _|
.
.   Figure 1.                         Figure 2.
.
The table contains 11 regions, see Figure 1.
The regions are distributed in 6 sections. The Figure 2 shows the sections separately.
Then consider the following table which contains the diagram of every region separately:
---------------------------------------------------------------------
|         |         |         |                    |       |        |
| Section | Region  |  Parts  |       Region       | Peri- |  a(n)  |
|         |         |(A220482)|       diagram      | meter |        |
---------------------------------------------------------------------
|         |         |         |      _             |       |        |
|    1    |    1    |    1    |     |_|            |   4   |    4   |
---------------------------------------------------------------------
|         |         |         |        _           |       |        |
|         |         |    1    |      _| |          |       |        |
|    2    |    2    |    2    |     |_ _|          |   8   |    8   |
---------------------------------------------------------------------
|         |         |         |          _         |       |        |
|         |         |    1    |         | |        |       |        |
|         |         |    1    |      _ _| |        |       |        |
|    3    |    3    |    3    |     |_ _ _|        |  12   |   12   |
---------------------------------------------------------------------
|         |         |         |      _ _           |       |        |
|         |    4    |    2    |     |_ _|          |   6   |        |
|         |---------|---------|----------------------------|        |
|         |         |         |            _       |       |        |
|         |         |    1    |           | |      |       |        |
|         |         |    1    |           | |      |       |        |
|         |         |    1    |          _| |      |       |        |
|         |         |    2    |      _ _|   |      |       |        |
|    4    |    5    |    4    |     |_ _ _ _|      |  18   |   24   |
---------------------------------------------------------------------
|         |         |         |      _ _ _         |       |        |
|         |    6    |    3    |     |_ _ _|        |   8   |        |
|         |---------|---------|--------------------|-------|        |
|         |         |         |              _     |       |        |
|         |         |    1    |             | |    |       |        |
|         |         |    1    |             | |    |       |        |
|         |         |    1    |             | |    |       |        |
|         |         |    1    |             | |    |       |        |
|         |         |    1    |            _| |    |       |        |
|         |         |    2    |      _ _ _|   |    |       |        |
|    5    |    7    |    5    |     |_ _ _ _ _|    |  24   |   32   |
---------------------------------------------------------------------
|         |         |         |      _ _           |       |        |
|         |    8    |    2    |     |_ _|          |   6   |        |
|         |---------|---------|--------------------|-------|        |
|         |         |         |          _ _       |       |        |
|         |         |    2    |      _ _|   |      |       |        |
|         |    9    |    4    |     |_ _ _ _|      |  12   |        |
1         |---------|---------|--------------------|-------|        |
|         |         |         |      _ _ _         |       |        |
|         |   10    |    3    |     |_ _ _|        |   8   |        |
|         |---------|---------|--------------------|-------|        |
|         |         |         |                _   |       |        |
|         |         |    1    |               | |  |       |        |
|         |         |    1    |               | |  |       |        |
|         |         |    1    |               | |  |       |        |
|         |         |    1    |               | |  |       |        |
|         |         |    1    |               | |  |       |        |
|         |         |    1    |               | |  |       |        |
|         |         |    1    |              _| |  |       |        |
|         |         |    2    |             |   |  |       |        |
|         |         |    2    |            _|   |  |       |        |
|         |         |    3    |      _ _ _|     |  |       |        |
|    6    |   11    |    6    |     |_ _ _ _ _ _|  |  34   |   60   |
---------------------------------------------------------------------
.
For n = 1..3, there is only one region in every section. The perimeters of the regions are 4, 8 and 12 respectively, so a(1) = 4, a(2) = 8, and a(3) = 12.
For n = 4, the 4th section contains two regions with perimeters 6 and 18 respectively. The sum of the perimeters is 6 + 18 = 24, so a(4) = 24.
For n = 5, the 5th section contains two regions with perimeters 8 and 24 respectively. The sum of the perimeters is 8 + 24 = 32, so a(5) = 32.
For n = 6, the 6th section contains four regions with perimeters 6, 12, 8 and 34 respectively. The sum of the perimeters is 6 + 12 + 8 + 34 = 60, so a(6) = 60.

Crossrefs

Partial sums give A278355.
Cf. A000041, A135010, A138137, A141285, A194446, A206437, A211992, A220482, A233968, A273140.

Formula

a(n) = 4 * A138137(n) = 2 * A233968(n), n >= 1 in both cases.

A299473 a(n) = 3*p(n), where p(n) is the number of partitions of n.

Original entry on oeis.org

3, 3, 6, 9, 15, 21, 33, 45, 66, 90, 126, 168, 231, 303, 405, 528, 693, 891, 1155, 1470, 1881, 2376, 3006, 3765, 4725, 5874, 7308, 9030, 11154, 13695, 16812, 20526, 25047, 30429, 36930, 44649, 53931, 64911, 78045, 93555, 112014, 133749, 159522, 189783, 225525, 267402, 316674, 374262, 441819, 520575, 612678
Offset: 0

Views

Author

Omar E. Pol, Feb 10 2018

Keywords

Comments

For n >= 1, a(n) is also the number of vertices in the minimalist diagram of partitions of n, in which A139582(n) is the number of line segments and A000041(n) is the number of open regions (see example).

Examples

			Construction of a minimalist version of a modular table of partitions in which a(n) is the number of vertices of the diagram after n-th stage (n = 1..6):
-----------------------------------------------------------------------------------
n.........:    1     2       3         4           5           6   (stage)
A000041(n):    1     2       3         5           7          11   (open regions)
A139582(n):    2     4       6        10          14          22   (line segments)
a(n)......:    3     6       9        15          21          33   (vertices)
-----------------------------------------------------------------------------------
r     p(n)
-----------------------------------------------------------------------------------
.
1 .... 1 .... _|   _| |   _| | |   _| | | |   _| | | | |   _| | | | | |
2 .... 2 ......... _ _|   _ _| |   _ _| | |   _ _| | | |   _ _| | | | |
3 .... 3 ................ _ _ _|   _ _ _| |   _ _ _| | |   _ _ _| | | |
4                                  _ _|   |   _ _|   | |   _ _|   | | |
5 .... 5 ......................... _ _ _ _|   _ _ _ _| |   _ _ _ _| | |
6                                             _ _ _|   |   _ _ _|   | |
7 .... 7 .................................... _ _ _ _ _|   _ _ _ _ _| |
8                                                          _ _|   |   |
9                                                          _ _ _ _|   |
10                                                         _ _ _|     |
11 .. 11 ................................................. _ _ _ _ _ _|
.
The r-th horizontal line segment has length A141285(r).
The r-th vertical line segment has length A194446(r).
An infinite diagram is a minimalist table of all partitions of all positive integers.

Links

Crossrefs

k times partition numbers: A000041 (k=1), A139582 (k=2), this sequence (k=3), A299474 (k=4).
Cf. A135010, A141285, A182181, A186114, A193870, A194446, A194447, A206437, A207779, A220482, A220517, A273140, A278355, A278602, A299475.

Formula

a(n) = 3*A000041(n) = A000041(n) + A139582(n).
a(n) = A299475(n) - 1, n >= 1.
Showing 1-4 of 4 results.

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