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%I #18 Sep 18 2021 21:54:49
%S 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,
%T 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,
%U 4,7,5,6,6,12,1,11,2,10,3,9,4,8,5,7,6,13,1,12,2,11
%N 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,... .
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F 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.
%F a(n) = A339399(A103889(n)). - _Wesley Ivan Hurt_, May 09 2021
%e [9,1]
%e [7,1] [8,1] [8,2]
%e [5,1] [6,1] [6,2] [7,2] [7,3]
%e [3,1] [4,1] [4,2] [5,2] [5,3] [6,3] [6,4]
%e [1,1] [2,1] [2,2] [3,2] [3,3] [4,3] [4,4] [5,4] [5,5]
%e k 2 3 4 5 6 7 8 9 10
%e --------------------------------------------------------------------------
%e k Nonincreasing partitions of k
%e --------------------------------------------------------------------------
%e 2 1,1
%e 3 2,1
%e 4 3,1,2,2
%e 5 4,1,3,2
%e 6 5,1,4,2,3,3
%e 7 6,1,5,2,4,3
%e 8 7,1,6,2,5,3,4,4
%e 9 8,1,7,2,6,3,5,4
%e 10 9,1,8,2,7,3,6,4,5,5
%e ...
%t 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}]
%Y Cf. A103889, A339399.
%Y Bisections: A199474, A122197.
%K nonn
%O 1,3
%A _Wesley Ivan Hurt_, Dec 05 2020
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