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reviewed
approved
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Revision History for A345023
(Bold, blue-underlined text is an addition; faded, red-underlined text is aShowing entries 1-10 | older changes
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.
(history; published version)
(history; published version)
STATUS
proposed
reviewed
STATUS
editing
proposed
EXAMPLE
For n = 7 we can see below some views of two associated polycubes called "prism of partitions" and "tower of divisors". Both objects contains the same number of cubes (that property is also valid for n >= 1).
Front view of the Partitions Position Lateral view of the
prism of partitions. of 7. of the 1's. of the tower of divisors.
Top view of the
of the tower of divisors.
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 of divisors that appears in the figures 4 and 5.
STATUS
approved
editing
STATUS
reviewed
approved
STATUS
proposed
reviewed
STATUS
editing
proposed
MATHEMATICA
Accumulate @ Table[4 * PartitionsP[k-1] + 2 * DivisorSigma[1, k + 1], {k, 0, 1, 50}] (* Amiram Eldar, Jul 14 2021 *)
STATUS
proposed
editing
STATUS
editing
proposed
MATHEMATICA
Accumulate @ Table[4 * PartitionsP[k] + 2 * DivisorSigma[1, k + 1], {k, 0, 50}] (* Amiram Eldar, Jul 14 2021 *)
STATUS
proposed
editing