A339443 - OEIS

The OEIS is supported by the many generous donors to the OEIS Foundation.

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give

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,... .

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 * A369323 A342655 A161901

Adjacent sequences: A339440 A339441 A339442 * A339444 A339445 A339446

KEYWORD

nonn

AUTHOR

Wesley Ivan Hurt, Dec 05 2020

STATUS

approved