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
Offset: 1
Keywords
Examples
[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
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k Nonincreasing partitions of k
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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
...
Programs
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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}]
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.