%I #14 Jan 27 2024 19:19:17

%S 0,0,0,1,3,10,27,68,156,357,775,1667,3505,7303,15019,30759,62489,

%T 126619,255542,514721,1034425,2076924,4164650,8346306,16715847,

%U 33467324,66982798,134040148,268179417,536510608,1073226084,2146759579,4293930436,8588485846,17177799658

%N 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 Andrew Howroyd, <a href="/A364534/b364534.txt">Table of n, a(n) for n = 0..85</a>

%F a(n) = 2^n - A151897(n). - _Andrew Howroyd_, Jan 27 2024

%e The a(0) = 0 through a(5) = 10 subsets:

%e . . . {1,2,3} {1,2,3} {1,2,3}

%e {1,3,4} {1,3,4}

%e {1,2,3,4} {1,4,5}

%e {2,3,5}

%e {1,2,3,4}

%e {1,2,3,5}

%e {1,2,4,5}

%e {1,3,4,5}

%e {2,3,4,5}

%e {1,2,3,4,5}

%t Table[Length[Select[Subsets[Range[n]],Intersection[#,Total/@Subsets[#,{2,Length[#]}]]!={}&]],{n,0,10}]

%Y The binary version is A088809, complement A085489.

%Y The complement is counted by A151897.

%Y The complement for partitions is A237667, ranks A364531.

%Y For partitions we have A237668, ranks A364532.

%Y For strict partitions we have A364272, complement A364349.

%Y A108917 counts knapsack partitions, strict A275972.

%Y A236912 counts sum-free partitions w/o re-used parts, complement A237113.

%Y Cf. A007865, A093971, A323092, A325862, A326083, A363225, A364345, A364346, A364348, A364350, A364533, A364670.

%K nonn

%O 0,5

%A _Gus Wiseman_, Aug 02 2023

%E a(16)-a(25) from _Chai Wah Wu_, Nov 14 2023

%E a(26) onwards (using A151897) added by _Andrew Howroyd_, Jan 27 2024