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.

0, 0, 0, 1, 3, 10, 27, 68, 156, 357, 775, 1667, 3505, 7303, 15019, 30759, 62489, 126619, 255542, 514721, 1034425, 2076924, 4164650, 8346306, 16715847, 33467324, 66982798, 134040148, 268179417, 536510608, 1073226084, 2146759579, 4293930436, 8588485846, 17177799658

Offset: 0

Author

Gus Wiseman, Aug 02 2023

Keywords

• nonn

Examples

			The a(0) = 0 through a(5) = 10 subsets:
  .  .  .  {1,2,3}  {1,2,3}    {1,2,3}
                    {1,3,4}    {1,3,4}
                    {1,2,3,4}  {1,4,5}
                               {2,3,5}
                               {1,2,3,4}
                               {1,2,3,5}
                               {1,2,4,5}
                               {1,3,4,5}
                               {2,3,4,5}
                               {1,2,3,4,5}
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..85

Crossrefs

The binary version is A088809, complement A085489.

The complement is counted by A151897.

The complement for partitions is A237667, ranks A364531.

For partitions we have A237668, ranks A364532.

For strict partitions we have A364272, complement A364349.

A108917 counts knapsack partitions, strict A275972.

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

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

Programs

Mathematica

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

Formula

a(n) = 2^n - A151897(n). - Andrew Howroyd, Jan 27 2024

Extensions

a(16)-a(25) from Chai Wah Wu, Nov 14 2023

a(26) onwards (using A151897) added by Andrew Howroyd, Jan 27 2024