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

0,5

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.

Links

Table of n, a(n) for n=0..15.

S. R. Finch, Monoids of natural numbers, March 17, 2009.

Formula

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

First differences of A364534.

Example

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}

Mathematica

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

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.

Sequence in context: A209398 A364756 A175660 * A347079 A370304 A175120

Adjacent sequences: A365066 A365067 A365068 * A365070 A365071 A365072

Keyword

nonn

Author

Gus Wiseman, Aug 26 2023

Status

approved