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A346533
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Irregular triangle read by rows in which row n lists the first n - 2 terms of A000203 together with the sum of A000203(n-1) and A000203(n), with a(1) = 1.
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5
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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
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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 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.
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LINKS
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EXAMPLE
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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
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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 *)
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CROSSREFS
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The length of row n is A028310(n-1).
Cf. A175254 (volume of the pyramid).
Cf. A000041, A221529, A237270, A237593, A245092, A245093 (similar), A336811, A336812, A338156, A339106, A340035.
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KEYWORD
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AUTHOR
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STATUS
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approved
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