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A364534 Number of subsets of {1..n} containing some element equal to the sum of two or more distinct other elements. A variation of sum-full subsets without re-used elements.

%I A364534 #14 Jan 27 2024 19:19:17
%S A364534 0,0,0,1,3,10,27,68,156,357,775,1667,3505,7303,15019,30759,62489,
%T A364534 126619,255542,514721,1034425,2076924,4164650,8346306,16715847,
%U A364534 33467324,66982798,134040148,268179417,536510608,1073226084,2146759579,4293930436,8588485846,17177799658
%N A364534 Number of subsets of {1..n} containing some element equal to the sum of two or more distinct other elements. A variation of sum-full subsets without re-used elements.
%H A364534 Andrew Howroyd, <a href="/A364534/b364534.txt">Table of n, a(n) for n = 0..85</a>
%F A364534 a(n) = 2^n - A151897(n). - _Andrew Howroyd_, Jan 27 2024
%e A364534 The a(0) = 0 through a(5) = 10 subsets:
%e A364534   .  .  .  {1,2,3}  {1,2,3}    {1,2,3}
%e A364534                     {1,3,4}    {1,3,4}
%e A364534                     {1,2,3,4}  {1,4,5}
%e A364534                                {2,3,5}
%e A364534                                {1,2,3,4}
%e A364534                                {1,2,3,5}
%e A364534                                {1,2,4,5}
%e A364534                                {1,3,4,5}
%e A364534                                {2,3,4,5}
%e A364534                                {1,2,3,4,5}
%t A364534 Table[Length[Select[Subsets[Range[n]],Intersection[#,Total/@Subsets[#,{2,Length[#]}]]!={}&]],{n,0,10}]
%Y A364534 The binary version is A088809, complement A085489.
%Y A364534 The complement is counted by A151897.
%Y A364534 The complement for partitions is A237667, ranks A364531.
%Y A364534 For partitions we have A237668, ranks A364532.
%Y A364534 For strict partitions we have A364272, complement A364349.
%Y A364534 A108917 counts knapsack partitions, strict A275972.
%Y A364534 A236912 counts sum-free partitions w/o re-used parts, complement A237113.
%Y A364534 Cf. A007865, A093971, A323092, A325862, A326083, A363225, A364345, A364346, A364348, A364350, A364533, A364670.
%K A364534 nonn
%O A364534 0,5
%A A364534 _Gus Wiseman_, Aug 02 2023
%E A364534 a(16)-a(25) from _Chai Wah Wu_, Nov 14 2023
%E A364534 a(26) onwards (using A151897) added by _Andrew Howroyd_, Jan 27 2024