A342913 Pairwise listing of the partitions of 2k into two parts, (s,t), with 0 < t <= s ordered by decreasing values of s and where k = 1,2,... .

1, 1, 3, 1, 2, 2, 5, 1, 4, 2, 3, 3, 7, 1, 6, 2, 5, 3, 4, 4, 9, 1, 8, 2, 7, 3, 6, 4, 5, 5, 11, 1, 10, 2, 9, 3, 8, 4, 7, 5, 6, 6, 13, 1, 12, 2, 11, 3, 10, 4, 9, 5, 8, 6, 7, 7, 15, 1, 14, 2, 13, 3, 12, 4, 11, 5, 10, 6, 9, 7, 8, 8, 17, 1, 16, 2, 15, 3, 14, 4, 13, 5, 12, 6, 11, 7

Offset

1,3

Links

Table of n, a(n) for n=1..86.

Index entries for sequences related to partitions

Formula

a(n) = k - (k^2 + k - m)*(-1)^n / 2, where k = round(sqrt(m)) and m = 2*floor((n+1-(-1)^n)/2).

a(n) = A342769(A103889(n)).

Example

                                                        [13,1]
                                               [11,1]   [12,2]
                                       [9,1]   [10,2]   [11,3]
                               [7,1]   [8,2]   [9, 3]   [10,4]
                       [5,1]   [6,2]   [7,3]   [8, 4]   [9, 5]
               [3,1]   [4,2]   [5,3]   [6,4]   [7, 5]   [8, 6]
       [1,1]   [2,2]   [3,3]   [4,4]   [5,5]   [6, 6]   [7, 7]
   2k    2       4       6       8       10      12       14
  --------------------------------------------------------------------------
   2k   Decreasing partitions of 2k
  --------------------------------------------------------------------------
   2   1,1
   4   3,1,2,2
   6   5,1,4,2,3,3
   8   7,1,6,2,5,3,4,4
  10   9,1,8,2,7,3,6,4,5,5
  12   11,1,10,2,9,3,8,4,7,5,6,6
  14   13,1,12,2,11,3,10,4,9,5,8,6,7,7
  ...

Crossrefs

Cf. A000194, A052928, A103889, A339399, A339443, A342769.

Sequence in context: A282016 A238848 A117494 * A221490 A231819 A286546

Adjacent sequences: A342910 A342911 A342912 * A342914 A342915 A342916

Keyword

nonn

Author

Wesley Ivan Hurt, Mar 28 2021

Status

approved