This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A232466 #52 Mar 21 2025 09:58:56
%S A232466 0,0,1,2,4,10,20,44,93,198,414,864,1788,3687,7541,15382,31200,63191,
%T A232466 127482,256857,516404,1037104,2080357,4170283,8354078,16728270,
%U A232466 33485553,67012082,134083661,268249350,536617010,1073391040,2147014212,4294321453,8589084469,17178702571,34358228044,68717407217,137436320023,274874294012,549751307200,1099505394507,2199015662477,4398035921221,8796080392378,17592168222674
%N A232466 Number of dependent sets with largest element n.
%C A232466 Let S be a set of positive integers. If S can be divided into two subsets which have equal sums, then S is said to be a dependent set.
%C A232466 Dependent sets are also called biquanimous sets. Biquanimous partitions are counted by A002219 and ranked by A357976. - _Gus Wiseman_, Apr 18 2024
%D A232466 J. Bourgain, Λ_p-sets in analysis: results, problems and related aspects. Handbook of the geometry of Banach spaces, Vol. I,195-232, North-Holland, Amsterdam, 2001.
%H A232466 Martin Fuller, <a href="/A232466/a232466.cpp.txt">C++ program</a>
%F A232466 a(n) < 2^(n-2) because there are 2^(n-1) sets of which half have an even sum. - _Martin Fuller_, Mar 21 2025
%e A232466 From _Gus Wiseman_, Apr 18 2024: (Start)
%e A232466 The a(1) = 0 through a(6) = 10 sets:
%e A232466 . . {1,2,3} {1,3,4} {1,4,5} {1,5,6}
%e A232466 {1,2,3,4} {2,3,5} {2,4,6}
%e A232466 {1,2,4,5} {1,2,3,6}
%e A232466 {2,3,4,5} {1,2,5,6}
%e A232466 {1,3,4,6}
%e A232466 {2,3,5,6}
%e A232466 {3,4,5,6}
%e A232466 {1,2,3,4,6}
%e A232466 {1,2,4,5,6}
%e A232466 {2,3,4,5,6}
%e A232466 (End)
%p A232466 b:= proc(n, i) option remember; `if`(i<1, `if`(n=0, {0}, {}),
%p A232466 `if`(i*(i+1)/2<n, {}, b(n, i-1) union map(p-> p+x^i,
%p A232466 b(n+i, i-1) union b(abs(n-i), i-1))))
%p A232466 end:
%p A232466 a:= n-> nops(b(n, n-1)):
%p A232466 seq(a(n), n=1..15); # _Alois P. Heinz_, Nov 24 2013
%t A232466 b[n_, i_] := b[n, i] = If[i<1, If[n == 0, {0}, {}], If[i*(i+1)/2 < n, {}, b[n, i-1] ~Union~ Map[Function[p, p+x^i], b[n+i, i-1] ~Union~ b[Abs[n-i], i-1]]]]; a[n_] := Length[b[n, n-1]]; Table[Print[a[n]]; a[n], {n, 1, 24}] (* _Jean-François Alcover_, Mar 04 2014, after _Alois P. Heinz_ *)
%t A232466 biqQ[y_]:=MemberQ[Total/@Subsets[y],Total[y]/2];
%t A232466 Table[Length[Select[Subsets[Range[n]], MemberQ[#,n]&&biqQ[#]&]],{n,10}] (* _Gus Wiseman_, Apr 18 2024 *)
%o A232466 (PARI) dep(S,k=0)=if(#S<2,return(if(#S,S[1],0)==k)); my(T=S[1..#S-1]);dep(T,abs(k-S[#S]))||dep(T,k+S[#S])
%o A232466 a(n)=my(S=[1..n-1]);sum(i=1,2^(n-1)-1,dep(vecextract(S,i),n)) \\ _Charles R Greathouse IV_, Nov 25 2013
%o A232466 (PARI) a(n)=my(r=0);forsubset(n-1,s,my(t=sum(i=1,#s,s[i])+n);if(t%2==0,my(b=1);for(i=1,#s,b=bitor(b,b<<s[i]));if(bittest(b,t\2),r++)));r; \\ _Martin Fuller_, Mar 21 2025
%Y A232466 Cf. A161943, A232534.
%Y A232466 Column k=2 of A248112.
%Y A232466 First differences of A371791.
%Y A232466 The complement is counted by A371793, differences of A371792.
%Y A232466 This is the "bi-" case of A371797, differences of A371796.
%Y A232466 A002219 (aerated) counts biquanimous partitions, ranks A357976.
%Y A232466 A006827 and A371795 count non-biquanimous partitions, ranks A371731.
%Y A232466 A237258 (aerated) counts biquanimous strict partitions, ranks A357854.
%Y A232466 A321142 and A371794 count non-biquanimous strict partitions.
%Y A232466 Cf. A035470, A064914, A321451, A321452, A366320, A367094, A371783, A371789.
%K A232466 nonn
%O A232466 1,4
%A A232466 _David S. Newman_, Nov 24 2013
%E A232466 a(9)-a(24) from _Alois P. Heinz_, Nov 24 2013
%E A232466 a(25) from _Alois P. Heinz_, Sep 30 2014
%E A232466 a(26) from _Alois P. Heinz_, Sep 17 2022
%E A232466 a(27) onwards from _Martin Fuller_, Mar 21 2025