A346875 Irregular triangle read by rows in which row n lists the row A000384(n) of A237591, n >= 1.

1, 4, 1, 1, 8, 3, 2, 1, 1, 15, 5, 3, 2, 1, 1, 1, 23, 8, 5, 2, 2, 2, 1, 1, 1, 34, 11, 6, 4, 3, 2, 2, 1, 1, 1, 1, 46, 16, 8, 5, 4, 2, 3, 1, 2, 1, 1, 1, 1, 61, 20, 11, 6, 5, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 77, 26, 14, 8, 5, 5, 3, 2, 3, 2, 1, 2, 1, 1, 1, 1, 1, 96, 32, 16

Offset

1,2

Comments

The characteristic shape of the symmetric representation of sigma(A000384(n)) consists in that in the main diagonal of the diagram the smallest Dyck path has a valley and the largest Dyck path has a peak.

So knowing this we can know if a number is a hexagonal number (or not) just by looking at the diagram, even ignoring the concept of hexagonal number.

Therefore we can see a geometric pattern of the distribution of the hexagonal numbers in the stepped pyramid described in A245092.

T(n,k) is also the length of the k-th line segment of the largest Dyck path of the symmetric representation of sigma(A000384(n-1)), from the border to the center, hence the sum of the n-th row of triangle is equal to A000384(n-1).

T(n,k) is also the difference between the total number of partitions of all positive integers <= n-th hexagonal number into exactly k consecutive parts, and the total number of partitions of all positive integers <= n-th hexagonal number into exactly k + 1 consecutive parts.

Links

Table of n, a(n) for n=1..84.

Example

Triangle begins:
   1;
   4,  1,  1;
   8,  3,  2,  1, 1;
  15,  5,  3,  2, 1, 1, 1;
  23,  8,  5,  2, 2, 2, 1, 1, 1;
  34, 11,  6,  4, 3, 2, 2, 1, 1, 1, 1;
  46, 16,  8,  5, 4, 2, 3, 1, 2, 1, 1, 1, 1;
  61, 20, 11,  6, 5, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1;
  77, 26, 14,  8, 5, 5, 3, 2, 3, 2, 1, 2, 1, 1, 1, 1, 1;
  96, 32, 16, 10, 7, 5, 4, 3, 3, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1;
...
Illustration of initial terms:
Column H gives the nonzero hexagonal numbers (A000384).
Column S gives the sum of the divisors of the hexagonal numbers which equals the area (and the number of cells) of the associated diagram.
-------------------------------------------------------------------------
  n    H    S   Diagram
-------------------------------------------------------------------------
                 _         _                 _                         _
  1    1    1   |_|       | |               | |                       | |
                 1        | |               | |                       | |
                       _ _| |               | |                       | |
                      |    _|               | |                       | |
                 _ _ _|  _|                 | |                       | |
  2    6   12   |_ _ _ _| 1                 | |                       | |
                    4    1                  | |                       | |
                                       _ _ _|_|                       | |
                                   _ _| |                             | |
                                  |    _|                             | |
                                 _|  _|                               | |
                                |_ _|1 1                              | |
                                | 2                                   | |
                 _ _ _ _ _ _ _ _|4                           _ _ _ _ _| |
  3   15   24   |_ _ _ _ _ _ _ _|                           |  _ _ _ _ _|
                        8                                   | |
                                                         _ _| |
                                                     _ _|  _ _|
                                                    |    _|
                                                   _|  _|
                                                  |  _|1 1
                                             _ _ _| | 1
                                            |  _ _ _|2
                                            | |  3
                                            | |
                                            | |5
                 _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
  4   28   56   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
                              15
.

Crossrefs

Row sums give A000384, n >= 1.

Row lengths give A005408.

Column 1 is A267682, n >= 1.

For the characteristic shape of sigma(A000040(n)) see A346871.

For the characteristic shape of sigma(A000079(n)) see A346872.

For the characteristic shape of sigma(A000217(n)) see A346873.

For the visualization of Mersenne numbers A000225 see A346874.

For the characteristic shape of sigma(A000396(n)) see A346876.

For the characteristic shape of sigma(A008588(n)) see A224613.

Cf. A000203, A237591, A237593, A245092, A249351, A262626.

Sequence in context: A119673 A144447 A051455 * A289511 A158687 A141541

Adjacent sequences: A346872 A346873 A346874 * A346876 A346877 A346878

Keyword

nonn,tabf

Author

Omar E. Pol, Aug 06 2021

Status

approved