OFFSET
1,2
COMMENTS
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 top of the symmetric tower (a polycube) described in A221529.
The height of the tower equals A000041(n-1).
The terraces of the tower 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 lower terrace of the tower is formed by two symmetric representations of sigma in the same level.
LINKS
Paolo Xausa, Table of n, a(n) for n = 1..11176 (rows 1..150 of the triangle, flattened)
EXAMPLE
Triangle begins:
1;
4;
1, 7;
1, 3, 11;
1, 3, 4, 13;
1, 3, 4, 7, 18;
1, 3, 4, 7, 6, 20;
1, 3, 4, 7, 6, 12, 23;
1, 3, 4, 7, 6, 12, 8, 28;
1, 3, 4, 7, 6, 12, 8, 15, 31;
1, 3, 4, 7, 6, 12, 8, 15, 13, 30;
1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 40;
1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 42;
1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 38;
...
For n = 7, sigma(7) = 1 + 7 = 8 and sigma(6) = 1 + 2 + 3 + 6 = 12, and 8 + 12 = 20, so the last term of row 7 is T(7,6) = 20. The other terms in row 7 are the first five terms of A000203, so the 7th row of the triangle is [1, 3, 4, 7, 6, 20].
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 the first two partition numbers A000041 are [1, 1]), so T(7,6) = sigma(7) + sigma(6) = 8 + 12 = 20.
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Illustration of initial terms:
Row 1 Row 2 Row 3 Row 4 Row 5 Row 6
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1 4 1 7 1 3 11 1 3 4 13 1 3 4 7 18
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MATHEMATICA
A346533row[n_]:=If[n==1, {1}, Join[DivisorSigma[1, Range[n-2]], {Total[DivisorSigma[1, {n-1, n}]]}]]; Array[A346533row, 15] (* Paolo Xausa, Oct 23 2023 *)
CROSSREFS
Mirror of A340584.
The length of row n is A028310(n-1).
Row sums give A024916.
Leading diagonal gives A092403.
Other diagonals give A000203.
Companion of A346562.
Cf. A175254 (volume of the pyramid).
Cf. A066186 (volume of the tower).
KEYWORD
AUTHOR
Omar E. Pol, Jul 22 2021
STATUS
approved