A365069 Number of subsets of {1..n} containing n and some element equal to the sum of two or more distinct other elements. A variation of non-binary sum-full subsets without re-usable elements.

0, 0, 0, 1, 2, 7, 17, 41, 88, 201, 418, 892, 1838, 3798, 7716, 15740

Offset: 0

Author

Gus Wiseman, Aug 26 2023

Keywords

• nonn

Comments

The complement is counted by A365071. The binary case is A364756. Allowing elements to be re-used gives A365070. A version for partitions (but not requiring n) is A237668.

Examples

			The subset {2,4,6} has 6 = 4 + 2 so is counted under a(6).
The subset {1,2,4,7} has 7 = 4 + 2 + 1 so is counted under a(7).
The subset {1,4,5,8} has 5 = 4 + 1 so is counted under a(8).
The a(0) = 0 through a(6) = 17 subsets:
  .  .  .  {1,2,3}  {1,3,4}    {1,4,5}      {1,5,6}
                    {1,2,3,4}  {2,3,5}      {2,4,6}
                               {1,2,3,5}    {1,2,3,6}
                               {1,2,4,5}    {1,2,4,6}
                               {1,3,4,5}    {1,2,5,6}
                               {2,3,4,5}    {1,3,4,6}
                               {1,2,3,4,5}  {1,3,5,6}
                                            {1,4,5,6}
                                            {2,3,4,6}
                                            {2,3,5,6}
                                            {2,4,5,6}
                                            {1,2,3,4,6}
                                            {1,2,3,5,6}
                                            {1,2,4,5,6}
                                            {1,3,4,5,6}
                                            {2,3,4,5,6}
                                            {1,2,3,4,5,6}
		

Links

• Steven R. Finch, Monoids of natural numbers, March 17, 2009.

Crossrefs

The complement w/ re-usable parts is A288728, first differences of A007865.

First differences of A364534.

The binary complement is A364755, first differences of A085489.

The binary version is A364756, first differences of A088809.

The version with re-usable parts is A365070, first differences of A093971.

The complement is counted by A365071, first differences of A151897.

A124506 counts nonnegative combination-free subsets, differences of A326083.

A365046 counts nonnegative combination-full subsets, differences of A364914.

For partitions: A108917, A236912, A237113, A237668, A364532, A364913.

Strict partitions: A116861, A364272, A364349, A364350, A364839, A364916.

Cf. A050291, A326080, A363226, A364346, A364348, A364670.

Programs

Mathematica

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

Formula

a(n) = 2^(n-1) - A365070(n).

First differences of A364534.