A371793 Number of non-biquanimous subsets of {1..n} containing n.

1, 2, 3, 6, 12, 22, 44, 84, 163, 314, 610, 1184, 2308, 4505, 8843, 17386, 34336, 67881, 134662, 267431, 532172, 1060048, 2113947, 4218325, 8423138, 16826162, 33623311, 67205646, 134351795, 268621562, 537124814, 1074092608, 2147953084, 4295613139, 8590784715, 17181035797, 34361248692, 68721546255, 137441586921, 274881519876, 549760320576, 1099517861045, 2199030848627, 4398057100987, 8796105652038, 17592203866158

Offset: 1

Author

Gus Wiseman, Apr 07 2024

Keywords

• nonn

Comments

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.

Examples

			The a(1) = 1 through a(5) = 12 subsets:
  {1}  {2}    {3}    {4}      {5}
       {1,2}  {1,3}  {1,4}    {1,5}
              {2,3}  {2,4}    {2,5}
                     {3,4}    {3,5}
                     {1,2,4}  {4,5}
                     {2,3,4}  {1,2,5}
                              {1,3,5}
                              {2,4,5}
                              {3,4,5}
                              {1,2,3,5}
                              {1,3,4,5}
                              {1,2,3,4,5}
		

Crossrefs

The complement is counted by A232466, differences of A371791.

This is the "bi-" version of A371790, differences of A371789.

First differences of A371792.

The complement is the "bi-" version of A371797, differences of A371796.

A002219 aerated counts biquanimous partitions, ranks A357976.

A006827 and A371795 count non-biquanimous partitions, ranks A371731.

A108917 counts knapsack partitions, ranks A299702, strict A275972.

A237258 aerated counts biquanimous strict partitions, ranks A357854.

A321142 and A371794 count non-biquanimous strict partitions.

A321451 counts non-quanimous partitions, ranks A321453.

A321452 counts quanimous partitions, ranks A321454.

A366754 counts non-knapsack partitions, ranks A299729, strict A316402.

A371737 counts quanimous strict partitions, complement A371736.

A371781 lists numbers with biquanimous prime signature, complement A371782.

A371783 counts k-quanimous partitions.

Cf. A035470, A064914, A365543, A365661, A365663, A366320, A365381, A367094, A371788.

Programs

Mathematica

biqQ[y_]:=MemberQ[Total/@Subsets[y],Total[y]/2];
Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&!biqQ[#]&]],{n,15}]

Extensions

a(16) onwards from Martin Fuller, Mar 21 2025