A342913 - OEIS

%I #9 Feb 12 2026 22:18:02

%S 1,1,3,1,2,2,5,1,4,2,3,3,7,1,6,2,5,3,4,4,9,1,8,2,7,3,6,4,5,5,11,1,10,

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

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

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

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%F a(n) = k - (k^2 + k - m)*(-1)^n / 2, where k = round(sqrt(m)) and m = 2*floor((n+1-(-1)^n)/2).

%F a(n) = A342769(A103889(n)).

%e                                                         [13,1]

%e                                                [11,1]   [12,2]

%e                                        [9,1]   [10,2]   [11,3]

%e                                [7,1]   [8,2]   [9, 3]   [10,4]

%e                        [5,1]   [6,2]   [7,3]   [8, 4]   [9, 5]

%e                [3,1]   [4,2]   [5,3]   [6,4]   [7, 5]   [8, 6]

%e        [1,1]   [2,2]   [3,3]   [4,4]   [5,5]   [6, 6]   [7, 7]

%e    2k    2       4       6       8       10      12       14

%e   --------------------------------------------------------------------------

%e    2k   Decreasing partitions of 2k

%e   --------------------------------------------------------------------------

%e    2   1,1

%e    4   3,1,2,2

%e    6   5,1,4,2,3,3

%e    8   7,1,6,2,5,3,4,4

%e   10   9,1,8,2,7,3,6,4,5,5

%e   12   11,1,10,2,9,3,8,4,7,5,6,6

%e   14   13,1,12,2,11,3,10,4,9,5,8,6,7,7

%e   ...

%o (Python)

%o from math import isqrt

%o def A342913(n): return (k:=(t:=isqrt(m:=(n+2 if n&1 else n)&-2))+(m>t*(t+1)))+((k*(k+1)-m if n&1 else m-k*(k+1))>>1) # _Chai Wah Wu_, Feb 11 2026

%Y Cf. A000194, A052928, A103889, A339399, A339443, A342769.

%K nonn

%O 1,3

%A _Wesley Ivan Hurt_, Mar 28 2021