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A278602 Sum of the perimeters of all regions of the n-th section of a modular table of partitions. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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.

LINKS

Table of n, a(n) for n=0..46.

FORMULA

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

EXAMPLE

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.

Sequence in context: A239053 A272708 A157416 * A059992 A050570 A233341

Adjacent sequences:  A278599 A278600 A278601 * A278603 A278604 A278605

KEYWORD

nonn

AUTHOR

Omar E. Pol, Nov 23 2016

STATUS

approved

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Last modified October 22 10:24 EDT 2019. Contains 328317 sequences. (Running on oeis4.)