cp's OEIS Frontend

A088809 Number of subsets of {1, ..., n} that are not sum-free.

%I A088809 #24 Feb 16 2025 08:32:51
%S A088809 0,0,0,1,3,10,27,67,154,350,763,1638,3450,7191,14831,30398,61891,
%T A088809 125557,253841,511818,1029863,2069341,4153060,8327646,16687483,
%U A088809 33422562,66916342,133936603,268026776,536277032,1072886163,2146245056,4293187682,8587371116
%N A088809 Number of subsets of {1, ..., n} that are not sum-free.
%C A088809 a(n) = 2^n - A085489(n); a non-sum-free subset contains at least one subset {u,v, w} with w=u+v.
%C A088809 A variation of binary sum-full sets where parts cannot be re-used, this sequence counts subsets of {1..n} with an element equal to the sum of two distinct others. The complement is counted by A085489. The non-binary version is A364534. For re-usable parts we have A093971. - _Gus Wiseman_, Aug 10 2023
%H A088809 Fausto A. C. Cariboni, <a href="/A088809/b088809.txt">Table of n, a(n) for n = 0..75</a>
%H A088809 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Sum-FreeSet.html">Sum-Free Set</a>
%H A088809 Reinhard Zumkeller, <a href="/A088808/a088808.txt">Illustration of initial terms</a>
%e A088809 From _Gus Wiseman_, Aug 10 2023: (Start)
%e A088809 The set S = {1,3,6,8} has pair-sums {4,7,9,11,14}, which are all missing from S, so it is not counted under a(8).
%e A088809 The set {1,4,6,7} has pair-sum 1 + 6 = 7, so is counted under a(7).
%e A088809 The a(1) = 0 through a(5) = 10 sets:
%e A088809   .  .  {1,2,3}  {1,2,3}    {1,2,3}
%e A088809                  {1,3,4}    {1,3,4}
%e A088809                  {1,2,3,4}  {1,4,5}
%e A088809                             {2,3,5}
%e A088809                             {1,2,3,4}
%e A088809                             {1,2,3,5}
%e A088809                             {1,2,4,5}
%e A088809                             {1,3,4,5}
%e A088809                             {2,3,4,5}
%e A088809                             {1,2,3,4,5}
%e A088809 (End)
%t A088809 Table[Length[Select[Subsets[Range[n]],Intersection[#,Total/@Subsets[#,{2}]]!={}&]],{n,0,10}] (* _Gus Wiseman_, Aug 10 2023 *)
%Y A088809 The complement is counted by A085489, differences A364755.
%Y A088809 With re-usable parts we have A093971, for partitions A363225.
%Y A088809 The complement for partitions is A236912:
%Y A088809   non-binary A237667,
%Y A088809   ranks A364461,
%Y A088809   strict A364533.
%Y A088809 The version for partitions is A237113:
%Y A088809   non-binary A237668,
%Y A088809   ranks A364462,
%Y A088809   strict A364670.
%Y A088809 The non-binary version is A364534, complement A151897.
%Y A088809 First differences are A364756.
%Y A088809 Cf. A000079, A007865, A050291, A051026, A103580, A288728, A326020, A326080, A326083, A364272, A364349.
%K A088809 nonn
%O A088809 0,5
%A A088809 _Reinhard Zumkeller_, Oct 19 2003
%E A088809 Terms a(32) and beyond from _Fausto A. C. Cariboni_, Sep 28 2020