A346864 Irregular triangle read by rows in which row n lists the row A014105(n) of A237591, n >= 1.

2, 1, 6, 2, 1, 1, 11, 4, 3, 1, 1, 1, 19, 6, 4, 2, 2, 1, 1, 1, 28, 10, 5, 3, 3, 2, 1, 1, 1, 1, 40, 13, 7, 5, 3, 2, 2, 2, 1, 1, 1, 1, 53, 18, 10, 5, 4, 3, 3, 2, 1, 2, 1, 1, 1, 1, 69, 23, 12, 7, 5, 4, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 86, 29, 15, 9, 6, 5, 4, 2, 3, 2, 2, 1, 2, 1, 1, 1, 1, 1

Offset: 1

Author

Omar E. Pol, Aug 17 2021

Keywords

• nonn
• tabf

Comments

The characteristic shape of the symmetric representation of sigma(A014105(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.

T(n,k) is also the length of the k-th line segment of the largest Dyck path of the symmetric representation of sigma(A014105(n)), from the border to the center, hence the sum of the n-th row of triangle is equal to A014105(n).

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

1 together with the first column gives A317186. - Michel Marcus, Jan 12 2025

Examples

			Triangle begins:
   2,  1;
   6,  2,  1, 1;
  11,  4,  3, 1, 1, 1;
  19,  6,  4, 2, 2, 1, 1, 1;
  28, 10,  5, 3, 3, 2, 1, 1, 1, 1;
  40, 13,  7, 5, 3, 2, 2, 2, 1, 1, 1, 1;
  53, 18, 10, 5, 4, 3, 3, 2, 1, 2, 1, 1, 1, 1;
  69, 23, 12, 7, 5, 4, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1;
  86, 29, 15, 9, 6, 5, 4, 2, 3, 2, 2, 1, 2, 1, 1, 1, 1, 1;
...
Illustration of initial terms:
Column h gives the n-th second hexagonal number (A014105).
Column S gives the sum of the divisors of the second hexagonal numbers which equals the area (and the number of cells) of the associated diagram.
--------------------------------------------------------------------------------------
  n   h   S   Diagram
--------------------------------------------------------------------------------------
                  _             _                     _                             _
                 | |           | |                   | |                           | |
              _ _|_|           | |                   | |                           | |
  1   3   4  |_ _|1            | |                   | |                           | |
               2               | |                   | |                           | |
                            _ _| |                   | |                           | |
                           |  _ _|                   | |                           | |
                        _ _|_|                       | |                           | |
                       |  _|1                        | |                           | |
              _ _ _ _ _| | 1                         | |                           | |
  2  10  18  |_ _ _ _ _ _|2                          | |                           | |
                   6                          _ _ _ _|_|                           | |
                                             | |                                   | |
                                            _| |                                   | |
                                           |  _|                                   | |
                                        _ _|_|                                     | |
                                    _ _|  _|1                                      | |
                                   |_ _ _|1 1                                      | |
                                   |  3                               _ _ _ _ _ _ _| |
                                   |4                                |    _ _ _ _ _ _|
              _ _ _ _ _ _ _ _ _ _ _|                                 |   |
  3  21  32  |_ _ _ _ _ _ _ _ _ _ _|                              _ _|   |
                       11                                        |       |
                                                                _|    _ _|
                                                               |     |
                                                            _ _|    _|
                                                        _ _|      _|
                                                       |        _|1
                                                  _ _ _|    _ _|1 1
                                                 |         | 2
                                                 |  _ _ _ _|2
                                                 | |   4
                                                 | |
                                                 | |6
                                                 | |
              _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
  4  36  91  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
                               19
.
		

Crossrefs

Row sums give A014105, n >= 1.

Row lengths give A005843.

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(A000384(n)) see A346875.

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

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

For the characteristic shape of sigma(A174973(n)) see A317305.

Cf. A000203, A237591, A237593, A245092, A249351, A262626, A317186.

Programs

PARI

row(n) = my(m=n*(2*n + 1)); vector((sqrtint(8*m+1)-1)\2, k, ceil((m+1)/k - (k+1)/2) - ceil((m+1)/(k+1) - (k+2)/2)); \\ Michel Marcus, Jan 12 2025