%I #81 Dec 12 2016 09:00:49
%S 0,4,8,12,24,32,60,76,128,168,256,332,496,628,896,1152,1580,2008,2716,
%T 3416,4528,5688,7388,9228,11872,14708,18684,23088,29004,35632,44440,
%U 54288,67168,81756,100384,121656,148552,179192,217556,261544,315836,378232,454748,542584,649500,772532,920912
%N Sum of the perimeters of all regions of the n-th section of a modular table of partitions.
%C 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.
%C 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.
%F a(n) = 4 * A138137(n) = 2 * A233968(n), n >= 1 in both cases.
%e 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):
%e |--------------|-----------------------------------------------------|
%e | Modular table| Sections |
%e | of partitions|-----------------------------------------------------|
%e | for n=1..6 | 1 2 3 4 5 6 |
%e 1--------------|-----------------------------------------------------|
%e . _ _ _ _ _ _ _ _ _ _ _ _
%e . |_| | | | | | |_| _| | | | | | | | | |
%e . |_ _| | | | | |_ _| _ _| | | | | | | |
%e . |_ _ _| | | | |_ _ _| _ _ _| | | | | |
%e . |_ _| | | | |_ _| | | | | |
%e . |_ _ _ _| | | |_ _ _ _| _ _ _ _| | | |
%e . |_ _ _| | | |_ _ _| | | |
%e . |_ _ _ _ _| | |_ _ _ _ _| _ _ _ _ _| |
%e . |_ _| | | |_ _| | |
%e . |_ _ _ _| | |_ _ _ _| |
%e . |_ _ _| | |_ _ _| |
%e . |_ _ _ _ _ _| |_ _ _ _ _ _|
%e .
%e . Figure 1. Figure 2.
%e .
%e The table contains 11 regions, see Figure 1.
%e The regions are distributed in 6 sections. The Figure 2 shows the sections separately.
%e Then consider the following table which contains the diagram of every region separately:
%e ---------------------------------------------------------------------
%e | | | | | | |
%e | Section | Region | Parts | Region | Peri- | a(n) |
%e | | |(A220482)| diagram | meter | |
%e ---------------------------------------------------------------------
%e | | | | _ | | |
%e | 1 | 1 | 1 | |_| | 4 | 4 |
%e ---------------------------------------------------------------------
%e | | | | _ | | |
%e | | | 1 | _| | | | |
%e | 2 | 2 | 2 | |_ _| | 8 | 8 |
%e ---------------------------------------------------------------------
%e | | | | _ | | |
%e | | | 1 | | | | | |
%e | | | 1 | _ _| | | | |
%e | 3 | 3 | 3 | |_ _ _| | 12 | 12 |
%e ---------------------------------------------------------------------
%e | | | | _ _ | | |
%e | | 4 | 2 | |_ _| | 6 | |
%e | |---------|---------|----------------------------| |
%e | | | | _ | | |
%e | | | 1 | | | | | |
%e | | | 1 | | | | | |
%e | | | 1 | _| | | | |
%e | | | 2 | _ _| | | | |
%e | 4 | 5 | 4 | |_ _ _ _| | 18 | 24 |
%e ---------------------------------------------------------------------
%e | | | | _ _ _ | | |
%e | | 6 | 3 | |_ _ _| | 8 | |
%e | |---------|---------|--------------------|-------| |
%e | | | | _ | | |
%e | | | 1 | | | | | |
%e | | | 1 | | | | | |
%e | | | 1 | | | | | |
%e | | | 1 | | | | | |
%e | | | 1 | _| | | | |
%e | | | 2 | _ _ _| | | | |
%e | 5 | 7 | 5 | |_ _ _ _ _| | 24 | 32 |
%e ---------------------------------------------------------------------
%e | | | | _ _ | | |
%e | | 8 | 2 | |_ _| | 6 | |
%e | |---------|---------|--------------------|-------| |
%e | | | | _ _ | | |
%e | | | 2 | _ _| | | | |
%e | | 9 | 4 | |_ _ _ _| | 12 | |
%e 1 |---------|---------|--------------------|-------| |
%e | | | | _ _ _ | | |
%e | | 10 | 3 | |_ _ _| | 8 | |
%e | |---------|---------|--------------------|-------| |
%e | | | | _ | | |
%e | | | 1 | | | | | |
%e | | | 1 | | | | | |
%e | | | 1 | | | | | |
%e | | | 1 | | | | | |
%e | | | 1 | | | | | |
%e | | | 1 | | | | | |
%e | | | 1 | _| | | | |
%e | | | 2 | | | | | |
%e | | | 2 | _| | | | |
%e | | | 3 | _ _ _| | | | |
%e | 6 | 11 | 6 | |_ _ _ _ _ _| | 34 | 60 |
%e ---------------------------------------------------------------------
%e .
%e 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.
%e 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.
%e 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.
%e 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.
%Y Partial sums give A278355.
%Y Cf. A000041, A135010, A138137, A141285, A194446, A206437, A211992, A220482, A233968, A273140.
%K nonn
%O 0,2
%A _Omar E. Pol_, Nov 23 2016
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