OFFSET
1,1
COMMENTS
The characteristic shape of the symmetric representation of a(n) consists in that in the main diagonal of the diagram the smallest Dyck path has a peak and the largest Dyck path has a valley.
So knowing this characteristic shape we can know if a number is a second hexagonal number (or not) just by looking at the diagram, even ignoring the concept of second hexagonal number.
Therefore we can see a geometric pattern of the distribution of the second hexagonal numbers in the stepped pyramid described in A245092.
EXAMPLE
a(3) = 32 because the sum of divisors of the third second hexagonal number (i.e., 21) is 1 + 3 + 7 + 21 = 32.
On the other hand we can see that in the main diagonal of every diagram the smallest Dyck path has a peak and the largest Dyck path has a valley as shown below.
Illustration of initial terms:
----------------------------------------------------------------------------
n h(n) a(n) Diagram
----------------------------------------------------------------------------
_ _ _
| | | | | |
_ _|_| | | | |
1 3 4 |_ _| | | | |
| | | |
_ _| | | |
| _ _| | |
_ _|_| | |
| _| | |
_ _ _ _ _| | | |
2 10 18 |_ _ _ _ _ _| | |
_ _ _ _|_|
| |
_| |
| _|
_ _|_|
_ _| _|
|_ _ _|
|
|
_ _ _ _ _ _ _ _ _ _ _| \
3 21 32 |_ _ _ _ _ _ _ _ _ _ _| _ _\
| \
_| _\
| | \
_ _| _| \
_ _| _|
| _|
_ _ _| _ _|
| |
| _ _ _ _|
| |
| |
| |
| |
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
4 36 91 |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
Column h gives the n-th second hexagonal number (A014105).
The widths of the main diagonal of the diagrams are [0, 0, 0, 1] respectively.
a(n) is the area (and the number of cells) of the n-th diagram.
For n = 3 the sum of the regions (or parts) of the third diagram is 11 + 5 + 5 + 11 = 32, so a(3) = 32.
For n = 4 the symmetric representation of a(4) = sigma(36) = 91 is partially illustrated because it is too big to include totally here.
MATHEMATICA
a[n_] := DivisorSigma[1, n*(2*n + 1)]; Array[a, 50] (* Amiram Eldar, Aug 18 2021 *)
PROG
(PARI) a(n) = sigma(n*(2*n + 1)); \\ Michel Marcus, Aug 18 2021
CROSSREFS
Bisection of A074285.
Some sequences that gives sum of divisors: A000225 (of powers of 2), A008864 (of prime numbers), A065764 (of squares), A073255 (of composites), A074285 (of triangular numbers, also of generalized hexagonal numbers), A139256 (of perfect numbers), A175926 (of cubes), A224613 (of multiples of 6), A346865 (of hexagonal numbers), A346867 (of numbers with middle divisors), A346868 (of numbers with no middle divisors), A347155 (of nontriangular numbers).
KEYWORD
nonn
AUTHOR
Omar E. Pol, Aug 17 2021
STATUS
approved