OFFSET
1,1
COMMENTS
The largest side of the base of the tower has length n.
The base of the tower is the symmetric representation of A024916(n).
The volume of the tower is equal to A066186(n).
The area of each lateral view of the tower is equal to A000070(n-1).
The above results are because the correspondence between divisors and partitions described in A338156 and A336812.
The tower is also a member of the family of the stepped pyramid described in A245092.
The equivalent sequence for the surface area of the stepped pyramid is A328366.
EXAMPLE
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).
_ _ _ _ _ _ _
|_ _ _ _ | 7
|_ _ _ _|_ | 4 3
|_ _ _ | | 5 2
|_ _ _|_ _|_ | 3 2 2 _
|_ _ _ | | 6 1 1 | |
|_ _ _|_ | | 3 3 1 1 | |
|_ _ | | | 4 2 1 1 | |
|_ _|_ _|_ | | 2 2 2 1 1 _|_|
|_ _ _ | | | 5 1 1 1 1 | |
|_ _ _|_ | | | 3 2 1 1 1 1 _|_ _|
|_ _ | | | | 4 1 1 1 1 1 1 | | |
|_ _|_ | | | | 2 2 1 1 1 1 1 1 _|_|_ _|
|_ _ | | | | | 3 1 1 1 1 1 1 1 1 _| |_ _ _|
|_ | | | | | | 2 1 1 1 1 1 1 1 1 1 1 _ _|_ _|_ _ _|
|_|_|_|_|_|_|_| 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |_ _|_|_ _ _ _|
.
Figure 1. Figure 2. Figure 3. Figure 4.
Front view of the Partitions Position Lateral view
prism of partitions. of 7. of the 1's. of the tower.
.
.
_ _ _ _ _ _ _
| | | | | |_| 1
| | | |_|_ _| 2
| |_|_ |_ _| 3
|_ _ |_ _ _| 4
|_ |_ _ _| 5
| | 6
|_ _ _ _| 7
.
Figure 5.
Top view
of the tower.
.
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.
Figure 2 shows the partitions of 7 in accordance with the diagram.
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.
MATHEMATICA
Accumulate @ Table[4 * PartitionsP[k-1] + 2 * DivisorSigma[1, k], {k, 1, 50}] (* Amiram Eldar, Jul 14 2021 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Omar E. Pol, Jun 05 2021
STATUS
approved