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A340584 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.

%I A340584 #101 Jul 25 2021 13:46:00
%S A340584 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,
%T A340584 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,
%U A340584 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
%N A340584 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.
%C A340584 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.
%e A340584 Triangle begins:
%e A340584    1;
%e A340584    4;
%e A340584    7,  1;
%e A340584   11,  3,  1;
%e A340584   13,  4,  3,  1;
%e A340584   18,  7,  4,  3,  1;
%e A340584   20,  6,  7,  4,  3,  1;
%e A340584   23, 12,  6,  7,  4,  3,  1;
%e A340584   28,  8, 12,  6,  7,  4,  3,  1;
%e A340584   31, 15,  8, 12,  6,  7,  4,  3,  1;
%e A340584   30, 13, 15,  8, 12,  6,  7,  4,  3,  1;
%e A340584   40, 18, 13, 15,  8, 12,  6,  7,  4,  3,  1;
%e A340584   42, 12, 18, 13, 15,  8, 12,  6,  7,  4,  3,  1;
%e A340584   38, 28, 12, 18, 13, 15,  8, 12,  6,  7,  4,  3,  1;
%e A340584 ...
%e A340584 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].
%e A340584 From _Omar E. Pol_, Jul 11 2021: (Start)
%e A340584 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 A340584                                            _
%e A340584                                           | |
%e A340584                                           | |
%e A340584                                           | |
%e A340584         _                                 |_|_
%e A340584        |_|_                               |   |
%e A340584        |_ _|_                             |_ _|_
%e A340584        |_ _|_|_                           |   | |
%e A340584        |_ _ _| |_                         |_ _|_|_
%e A340584        |_ _ _|_ _|_                       |_ _ _| |_
%e A340584        |_ _ _ _| | |_                     |_ _ _|_ _|_ _
%e A340584        |_ _ _ _|_|_ _|                    |_ _ _ _|_|_ _|
%e A340584 .
%e A340584          Figure 1.                           Figure 2.
%e A340584         Lateral view                       Lateral view
%e A340584        of the pyramid.                     of the tower.
%e A340584 .
%e A340584 .       _ _ _ _ _ _ _                      _ _ _ _ _ _ _
%e A340584        |_| | | | | | |                    |_| | | | |   |
%e A340584        |_ _|_| | | | |                    |_ _|_| | |   |
%e A340584        |_ _|  _|_| | |                    |_ _|  _|_|   |
%e A340584        |_ _ _|    _|_|                    |_ _ _|    _ _|
%e A340584        |_ _ _|  _|                        |_ _ _|  _|
%e A340584        |_ _ _ _|                          |       |
%e A340584        |_ _ _ _|                          |_ _ _ _|
%e A340584 .
%e A340584           Figure 3.                          Figure 4.
%e A340584           Top view                           Top view
%e A340584        of the pyramid.                     of the tower.
%e A340584 .
%e A340584 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 A340584 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)
%t A340584 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 *)
%Y A340584 The length of row n is A028310(n-1).
%Y A340584 Row sums give A024916.
%Y A340584 Column 1 gives 1 together with A092403.
%Y A340584 Other columns give A000203.
%Y A340584 Cf. A175254 (volume of the pyramid).
%Y A340584 Cf. A066186 (volume of the tower).
%Y A340584 Cf. A346533 (mirror).
%Y A340584 Cf. A000041, A221529, A237270, A237593, A245092, A272172 (similar), A336811, A338156, A339106, A340035.
%K A340584 nonn,tabf
%O A340584 1,2
%A A340584 _Omar E. Pol_, Jan 12 2021