This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A345023 #45 Jun 09 2022 08:47:59
%S A345023 6,16,32,58,90,142,202,292,406,562,754,1034,1370,1822,2410,3176,4136,
%T A345023 5402,6982,9026,11598,14838,18894,24034,30396,38312,48136,60288,75220,
%U A345023 93624,116104,143598,177090,217770,267106,326820,398804,485472,589644,714564,864000,1042524,1255308
%N A345023 a(n) is the surface area of the symmetric tower described in A221529 which is a polycube whose successive terraces are the symmetric representation of sigma A000203(i) (from i = 1 to n) starting from the top and the levels of these terraces are the partition numbers A000041(h-1) (from h = 1 to n) starting from the base.
%C A345023 The largest side of the base of the tower has length n.
%C A345023 The base of the tower is the symmetric representation of A024916(n).
%C A345023 The volume of the tower is equal to A066186(n).
%C A345023 The area of each lateral view of the tower is equal to A000070(n-1).
%C A345023 The growth of the volume of the tower represents the convolution of A000203 and A000041.
%C A345023 The above results are because the correspondence between divisors and partitions described in A338156 and A336812.
%C A345023 The tower is also a member of the family of the stepped pyramid described in A245092.
%C A345023 The equivalent sequence for the surface area of the stepped pyramid is A328366.
%F A345023 a(n) = 4*A000070(n-1) + 2*A024916(n).
%F A345023 a(n) = 4*A000070(n-1) + A327329(n).
%e A345023 For n = 7 we can see below some views of two associated polycubes called "prism of partitions" and "tower". Both objects contains the same number of cubes (that property is also valid for n >= 1).
%e A345023 _ _ _ _ _ _ _
%e A345023 |_ _ _ _ | 7
%e A345023 |_ _ _ _|_ | 4 3
%e A345023 |_ _ _ | | 5 2
%e A345023 |_ _ _|_ _|_ | 3 2 2 _
%e A345023 |_ _ _ | | 6 1 1 | |
%e A345023 |_ _ _|_ | | 3 3 1 1 | |
%e A345023 |_ _ | | | 4 2 1 1 | |
%e A345023 |_ _|_ _|_ | | 2 2 2 1 1 _|_|
%e A345023 |_ _ _ | | | 5 1 1 1 1 | |
%e A345023 |_ _ _|_ | | | 3 2 1 1 1 1 _|_ _|
%e A345023 |_ _ | | | | 4 1 1 1 1 1 1 | | |
%e A345023 |_ _|_ | | | | 2 2 1 1 1 1 1 1 _|_|_ _|
%e A345023 |_ _ | | | | | 3 1 1 1 1 1 1 1 1 _| |_ _ _|
%e A345023 |_ | | | | | | 2 1 1 1 1 1 1 1 1 1 1 _ _|_ _|_ _ _|
%e A345023 |_|_|_|_|_|_|_| 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |_ _|_|_ _ _ _|
%e A345023 .
%e A345023 Figure 1. Figure 2. Figure 3. Figure 4.
%e A345023 Front view of the Partitions Position Lateral view
%e A345023 prism of partitions. of 7. of the 1's. of the tower.
%e A345023 .
%e A345023 .
%e A345023 _ _ _ _ _ _ _
%e A345023 | | | | | |_| 1
%e A345023 | | | |_|_ _| 2
%e A345023 | |_|_ |_ _| 3
%e A345023 |_ _ |_ _ _| 4
%e A345023 |_ |_ _ _| 5
%e A345023 | | 6
%e A345023 |_ _ _ _| 7
%e A345023 .
%e A345023 Figure 5.
%e A345023 Top view
%e A345023 of the tower.
%e A345023 .
%e A345023 Figure 1 is a two-dimensional diagram of the partitions of 7. The area of the diagram is A066186(7) = 105. Note that the diagram can be interpreted also as the front view of a right prism whose volumen is 1*7*A000041(7) = 1*7*15 = 105, equaling the volume of the tower that appears in the figures 4 and 5.
%e A345023 Figure 2 shows the partitions of 7 in accordance with the diagram.
%e A345023 Note that the shape and the area of the lateral view of the tower are the same as the shape and the area where the 1's are located in the diagram of partitions, see the figures 3 and 4. In this case the mentioned area equals A000070(7-1) = 30.
%e A345023 The connection between these two objects is a representation of the correspondence divisor/part described in A338156. See also A336812.
%t A345023 Accumulate @ Table[4 * PartitionsP[k-1] + 2 * DivisorSigma[1, k], {k, 1, 50}] (* _Amiram Eldar_, Jul 14 2021 *)
%Y A345023 Cf. A000041, A000070, A000203, A024916, A066186, A176206, A196020, A221529, A236104, A237593, A244050, A245092, A327329, A328366, A336811, A336812, A338156.
%K A345023 nonn
%O A345023 1,1
%A A345023 _Omar E. Pol_, Jun 05 2021