A352446 Total number of parts in all partitions of n into an even number of consecutive parts.

0, 0, 2, 0, 2, 0, 2, 0, 2, 4, 2, 0, 2, 4, 2, 0, 2, 4, 2, 0, 8, 4, 2, 0, 2, 4, 8, 0, 2, 4, 2, 0, 8, 4, 2, 8, 2, 4, 8, 0, 2, 4, 2, 8, 8, 4, 2, 0, 2, 4, 8, 8, 2, 4, 12, 0, 8, 4, 2, 8, 2, 4, 8, 0, 12, 4, 2, 8, 8, 4, 2, 0, 2, 4, 18, 8, 2, 16, 2, 0, 8, 4, 2, 8, 12, 4, 8, 0, 2, 16, 2

Offset: 1

Author

Omar E. Pol, Mar 16 2022

Keywords

• nonn

Examples

			For n = 21  the partitions of 21 into an even number of consecutive parts are [11, 10] and [6, 5, 4, 3, 2, 1]. The total number of parts in these two partitions is equal to 2 + 6 = 8, so a(21) = 8.
On the other hand consider the diagram below which is formed by the even-indexed staircase walks from the diagram of A286000.
The diagram is infinite and we have that:
The m-th staircase walk starts at row A014105(m).
The number of horizontal line segment in the n-th row equals A131576(n), the number of partitions of n into an even number of consecutive parts.
a(n) is the total length of all vertical line segments that are below and that share one vertex with the horizontal line segments that are in the n-th level of the diagram.
---------------------------------------------
   n    a(n)              Diagram
---------------------------------------------
   1      0
   2      0                               _
   3      2                              |2
   4      0                             _|1
   5      2                            |3
   6      0                           _|2
   7      2                          |4
   8      0                         _|3
   9      2                        |5     _
  10      4                       _|4    |4
  11      2                      |6      |3
  12      0                     _|5      |2
  13      2                    |7       _|1
  14      4                   _|6      |5
  15      2                  |8        |4
  16      0                 _|7        |3
  17      2                |9         _|2
  18      4               _|8        |6
  19      2              |10         |5
  20      0             _|9          |4   _
  21      8            |11          _|3  |6
  22      4           _|10         |7    |5
  23      2          |12           |6    |4
  24      0         _|11           |5    |3
  25      2        |13            _|4    |2
  26      4       _|12           |8     _|1
  27      8      |14             |7    |7
  28      0      |13             |6    |6
...
For n = 21 the number of horizontal line segment in the 21th row of the diagram equals A131576(21) = 2, the number of partitions of 21 into an even number of consecutive parts.
The total length of all vertical line segments that are below and that share one vertex with the horizontal line segments that are in the 21-th level of the diagram is equal to 2 + 6 = 8, so a(21) = 8.
		

Crossrefs

For more information about the diagram see A286000 and A237593.

Cf. A014105, A131576, A204217, A299765, A341309, A352505.

Formula

a(n) = A204217(n) - A341309(n), conjectured.

Conjecture: G.f.: Sum_{k>=1} 2*k*x^(k*(2*k+1))/(1-x^(2*k)). - Vaclav Kotesovec, Oct 23 2024