A342769 Pairwise listing of the partitions of 2k into two parts, (s,t), with 0 < s <= t ordered by increasing values of s and where k = 1,2,... .

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

Offset

1,4

Links

Table of n, a(n) for n=1..85.

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)/2).

a(n) = A342913(A103889(n)). - Wesley Ivan Hurt, May 09 2021

Example

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

Prog

(Python)
from math import isqrt
def A342769(n): return (k:=(t:=isqrt(m:=n+1&-2))+(m>t*(t+1)))+((m-k*(k+1) if n&1 else k*(k+1)-m)>>1) # Chai Wah Wu, Feb 11 2026

Crossrefs

Cf. A000194, A052928, A103889, A339399, A339443, A342913.

Sequence in context: A239066 A239067 A131015 * A130195 A071048 A357104

Adjacent sequences: A342766 A342767 A342768 * A342770 A342771 A342772

Keyword

nonn,tabf

Author

Wesley Ivan Hurt, Mar 21 2021

Status

approved