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A339443 Pairwise listing of the partitions of k into two parts (s,t), with 0 < t <= s ordered by decreasing values of s and where k = 2,3,... . 5
1, 1, 2, 1, 3, 1, 2, 2, 4, 1, 3, 2, 5, 1, 4, 2, 3, 3, 6, 1, 5, 2, 4, 3, 7, 1, 6, 2, 5, 3, 4, 4, 8, 1, 7, 2, 6, 3, 5, 4, 9, 1, 8, 2, 7, 3, 6, 4, 5, 5, 10, 1, 9, 2, 8, 3, 7, 4, 6, 5, 11, 1, 10, 2, 9, 3, 8, 4, 7, 5, 6, 6, 12, 1, 11, 2, 10, 3, 9, 4, 8, 5, 7, 6, 13, 1, 12, 2, 11 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

Table of n, a(n) for n=1..89.

Index entries for sequences related to partitions

FORMULA

a(n) = (1-(-1)^n)*(1+floor(sqrt(2*n-1)))/2-(((-1)^n-2*n-1)/2 + 2*Sum_{k=1..-1+floor(sqrt(2*n-2-(-1)^n))} floor((k+1)/2))*(-1)^n/2.

a(n) = A339399(A103889(n)). - Wesley Ivan Hurt, May 09 2021

EXAMPLE

                                                                     [9,1]

                                                     [7,1]   [8,1]   [8,2]

                                     [5,1]   [6,1]   [6,2]   [7,2]   [7,3]

                     [3,1]   [4,1]   [4,2]   [5,2]   [5,3]   [6,3]   [6,4]

     [1,1]   [2,1]   [2,2]   [3,2]   [3,3]   [4,3]   [4,4]   [5,4]   [5,5]

   k   2       3       4       5       6       7       8       9      10

  --------------------------------------------------------------------------

   k   Nonincreasing partitions of k

  --------------------------------------------------------------------------

   2   1,1

   3   2,1

   4   3,1,2,2

   5   4,1,3,2

   6   5,1,4,2,3,3

   7   6,1,5,2,4,3

   8   7,1,6,2,5,3,4,4

   9   8,1,7,2,6,3,5,4

  10   9,1,8,2,7,3,6,4,5,5

  ...

MATHEMATICA

Table[(1 - (-1)^n) (1 + Floor[Sqrt[2 n - 1]])/2 - (((-1)^n - 2 n - 1)/2 + 2 Sum[Floor[(k + 1)/2], {k, -1 + Floor[Sqrt[2 n - 2 - (-1)^n]]}]) (-1)^n/2, {n, 100}]

CROSSREFS

Cf. A103889, A339399.

Bisections: A199474, A122197.

Sequence in context: A227872 A323165 A091948 * A342655 A161901 A102862

Adjacent sequences:  A339440 A339441 A339442 * A339444 A339445 A339446

KEYWORD

nonn

AUTHOR

Wesley Ivan Hurt, Dec 05 2020

STATUS

approved

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Last modified October 22 18:11 EDT 2021. Contains 348175 sequences. (Running on oeis4.)