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A340584
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Irregular triangle read by rows T(n,k) in which row n lists sigma(n) + sigma(n-1) together with the first n - 2 terms of A000203 in reverse order, with T(1,1) = 1, n >= 1.
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6
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1, 4, 7, 1, 11, 3, 1, 13, 4, 3, 1, 18, 7, 4, 3, 1, 20, 6, 7, 4, 3, 1, 23, 12, 6, 7, 4, 3, 1, 28, 8, 12, 6, 7, 4, 3, 1, 31, 15, 8, 12, 6, 7, 4, 3, 1, 30, 13, 15, 8, 12, 6, 7, 4, 3, 1, 40, 18, 13, 15, 8, 12, 6, 7, 4, 3, 1, 42, 12, 18, 13, 15, 8, 12, 6, 7, 4, 3, 1, 38, 28, 12, 18, 13, 15, 8, 12, 6, 7, 4, 3, 1
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OFFSET
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1,2
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COMMENTS
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T(n,k) is the total area (or number of cells) of the terraces that are in the k-th level that contains terraces starting from the base of the symmetric tower (a polycube) described in A221529 which has A000041(n-1) levels in total. The terraces of the polycube are the symmetric representation of sigma. The terraces are in the levels that are the partition numbers A000041 starting from the base. Note that for n >= 2 there are n - 1 terraces because the first terrace of the tower is formed by two symmetric representations of sigma in the same level. The volume (or the number of cubes) equals A066186(n), the sum of all parts of all partitions of n. The volume is also the sum of all divisors of all terms of the first n rows of A336811. That is due to the correspondence between divisors and partitions (cf. A336811). The growth of the volume (A066186) represents the convolution of A000203 and A000041.
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LINKS
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EXAMPLE
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Triangle begins:
1;
4;
7, 1;
11, 3, 1;
13, 4, 3, 1;
18, 7, 4, 3, 1;
20, 6, 7, 4, 3, 1;
23, 12, 6, 7, 4, 3, 1;
28, 8, 12, 6, 7, 4, 3, 1;
31, 15, 8, 12, 6, 7, 4, 3, 1;
30, 13, 15, 8, 12, 6, 7, 4, 3, 1;
40, 18, 13, 15, 8, 12, 6, 7, 4, 3, 1;
42, 12, 18, 13, 15, 8, 12, 6, 7, 4, 3, 1;
38, 28, 12, 18, 13, 15, 8, 12, 6, 7, 4, 3, 1;
...
For n = 7, sigma(7) = 1 + 7 = 8 and sigma(6) = 1 + 2 + 3 + 6 = 12, and 8 + 12 = 20, so the first term of row 7 is T(7,1) = 20. The other terms in row 7 are the first five terms of A000203 in reverse order, that is [6, 7, 4, 3, 1] so the 7th row of the triangle is [20, 6, 7, 4, 3, 1].
For n = 7 we can see below the top view and the lateral view of the pyramid described in A245092 (with seven levels) and the top view and the lateral view of the tower described in A221529 (with 11 levels).
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Figure 1. Figure 2.
Lateral view Lateral view
of the pyramid. of the tower.
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Figure 3. Figure 4.
Top view Top view
of the pyramid. of the tower.
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Both polycubes have the same base which has an area equal to A024916(7) = 41 equaling the sum of the 7th row of triangle.
Note that in the top view of the tower the symmetric representation of sigma(6) and the symmetric representation of sigma(7) appear unified in the level 1 of the structure as shown above in the figure 4 (that is due to the first two partition numbers A000041 are [1, 1]), so T(7,1) = sigma(7) + sigma(6) = 8 + 12 = 20. (End)
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MATHEMATICA
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Table[If[n <= 2, {Total@ #}, Prepend[#2, Total@ #1] & @@ TakeDrop[#, 2]] &@ DivisorSigma[1, Range[n, 1, -1]], {n, 14}] // Flatten (* Michael De Vlieger, Jan 13 2021 *)
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CROSSREFS
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The length of row n is A028310(n-1).
Column 1 gives 1 together with A092403.
Cf. A175254 (volume of the pyramid).
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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