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%I A346533 #57 Jul 19 2024 12:04:13
%S A346533 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,
%T A346533 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,
%U A346533 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
%N A346533 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.
%C A346533 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.
%C A346533 The height of the tower equals A000041(n-1).
%C A346533 The terraces of the tower are the symmetric representation of sigma.
%C A346533 The terraces are in the levels that are the partition numbers A000041 starting from the base.
%C A346533 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.
%H A346533 Paolo Xausa, <a href="/A346533/b346533.txt">Table of n, a(n) for n = 1..11176</a> (rows 1..150 of the triangle, flattened)
%H A346533 Omar E. Pol, <a href="/A066186/a066186_1.jpg">Illustration of the partitions of 10, prism and tower</a>
%e A346533 Triangle begins:
%e A346533 1;
%e A346533 4;
%e A346533 1, 7;
%e A346533 1, 3, 11;
%e A346533 1, 3, 4, 13;
%e A346533 1, 3, 4, 7, 18;
%e A346533 1, 3, 4, 7, 6, 20;
%e A346533 1, 3, 4, 7, 6, 12, 23;
%e A346533 1, 3, 4, 7, 6, 12, 8, 28;
%e A346533 1, 3, 4, 7, 6, 12, 8, 15, 31;
%e A346533 1, 3, 4, 7, 6, 12, 8, 15, 13, 30;
%e A346533 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 40;
%e A346533 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 42;
%e A346533 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 38;
%e A346533 ...
%e A346533 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].
%e A346533 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).
%e A346533 _
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%e A346533 .
%e A346533 Figure 1. Figure 2.
%e A346533 Lateral view Lateral view
%e A346533 of the pyramid. of the tower.
%e A346533 .
%e A346533 . _ _ _ _ _ _ _ _ _ _ _ _ _ _
%e A346533 |_| | | | | | | |_| | | | | |
%e A346533 |_ _|_| | | | | |_ _|_| | | |
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%e A346533 |_ _ _| _|_| |_ _ _| _ _|
%e A346533 |_ _ _| _| |_ _ _| _|
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%e A346533 |_ _ _ _| |_ _ _ _|
%e A346533 .
%e A346533 Figure 3. Figure 4.
%e A346533 Top view Top view
%e A346533 of the pyramid. of the tower.
%e A346533 .
%e A346533 Both polycubes have the same base which has an area equal to A024916(7) = 41 equaling the sum of the 7th row of triangle.
%e A346533 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.
%e A346533 .
%e A346533 Illustration of initial terms:
%e A346533 Row 1 Row 2 Row 3 Row 4 Row 5 Row 6
%e A346533 .
%e A346533 1 4 1 7 1 3 11 1 3 4 13 1 3 4 7 18
%e A346533 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
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%e A346533 |_ _| | _| |_ _| | |_ _|_| | |_ _|_| | |
%e A346533 |_ _| | _| |_ _| _ _| |_ _| _| |
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%e A346533 |_ _ _ _|
%e A346533 .
%t A346533 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 *)
%Y A346533 Mirror of A340584.
%Y A346533 The length of row n is A028310(n-1).
%Y A346533 Row sums give A024916.
%Y A346533 Leading diagonal gives A092403.
%Y A346533 Other diagonals give A000203.
%Y A346533 Companion of A346562.
%Y A346533 Cf. A175254 (volume of the pyramid).
%Y A346533 Cf. A066186 (volume of the tower).
%Y A346533 Cf. A000041, A221529, A237270, A237593, A245092, A245093 (similar), A336811, A336812, A338156, A339106, A340035.
%K A346533 nonn,look,tabf
%O A346533 1,2
%A A346533 _Omar E. Pol_, Jul 22 2021