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A371792 Number of non-biquanimous subsets of {1..n}. Sets with no subset having the same sum as the complement.

%I A371792 #9 Mar 21 2025 09:53:12
%S A371792 0,1,3,6,12,24,46,90,174,337,651,1261,2445,4753,9258,18101,35487,
%T A371792 69823,137704,272366,539797,1071969,2132017,4245964,8464289,16887427,
%U A371792 33713589,67336900,134542546,268894341,537515903,1074640717,2148733325,4296686409,8592299548,17183084263,34364120060,68725368752,137446915007,274888501928,549770021804,1099530342380,2199048203425,4398079052052,8796136153039,17592241805077,35184445671235
%N A371792 Number of non-biquanimous subsets of {1..n}. Sets with no subset having the same sum as the complement.
%C A371792 A finite multiset of numbers is defined to be biquanimous iff it can be partitioned into two multisets with equal sums. Biquanimous partitions are counted by A002219 and ranked by A357976.
%e A371792 The subsets of S = {1,4,6,7} have distinct sums {0,1,4,5,6,7,8,10,11,12,13,14,17,18}. Since 9 is missing, S is counted under a(7).
%e A371792 The a(0) = 0 through a(4) = 12 subsets:
%e A371792   .  {1}  {1}    {1}    {1}
%e A371792           {2}    {2}    {2}
%e A371792           {1,2}  {3}    {3}
%e A371792                  {1,2}  {4}
%e A371792                  {1,3}  {1,2}
%e A371792                  {2,3}  {1,3}
%e A371792                         {1,4}
%e A371792                         {2,3}
%e A371792                         {2,4}
%e A371792                         {3,4}
%e A371792                         {1,2,4}
%e A371792                         {2,3,4}
%t A371792 biqQ[y_]:=MemberQ[Total/@Subsets[y],Total[y]/2];
%t A371792 Table[Length[Select[Subsets[Range[n]],Not@*biqQ]],{n,0,10}]
%Y A371792 This is the "bi-" version of A371789, differences A371790.
%Y A371792 The complement is counted by A371791, differences A232466.
%Y A371792 First differences are A371793.
%Y A371792 The complement is the "bi-" version of A371796, differences A371797.
%Y A371792 A002219 aerated counts biquanimous partitions, ranks A357976.
%Y A371792 A006827 and A371795 count non-biquanimous partitions, ranks A371731.
%Y A371792 A108917 counts knapsack partitions, ranks A299702, strict A275972.
%Y A371792 A237258 aerated counts biquanimous strict partitions, ranks A357854.
%Y A371792 A321142 and A371794 count non-biquanimous strict partitions.
%Y A371792 A321451 counts non-quanimous partitions, ranks A321453.
%Y A371792 A321452 counts quanimous partitions, ranks A321454.
%Y A371792 A366754 counts non-knapsack partitions, ranks A299729, strict A316402.
%Y A371792 A371737 counts quanimous strict partitions, complement A371736.
%Y A371792 A371781 lists numbers with biquanimous prime signature, complement A371782.
%Y A371792 A371783 counts k-quanimous partitions.
%Y A371792 Cf. A035470, A064914, A318434, A321455, A365543, A365661, A365663, A366320, A365381, A365925, A367094, A371788.
%K A371792 nonn
%O A371792 0,3
%A A371792 _Gus Wiseman_, Apr 07 2024
%E A371792 a(16) onwards from _Martin Fuller_, Mar 21 2025