A365070 Number of subsets of {1..n} containing n and some element equal to the sum of two other (possibly equal) elements.

0, 0, 1, 1, 5, 9, 24, 46, 109, 209, 469, 922, 1932, 3858, 7952, 15831, 32214, 64351, 129813, 259566, 521681, 1042703, 2091626, 4182470, 8376007, 16752524, 33530042, 67055129, 134165194, 268328011, 536763582, 1073523097, 2147268041, 4294505929, 8589506814, 17178978145

Offset: 0

Author

Gus Wiseman, Aug 24 2023

Keywords

• nonn

Comments

These are binary sum-full sets where elements can be re-used. The complement is counted by A288728. The non-binary version is A365046, complement A124506. For non-re-usable parts we have A364756, complement A085489.

Examples

			The subset {1,3} has no element equal to the sum of two others, so is not counted under a(3).
The subset {3,4,5} has no element equal to the sum of two others, so is not counted under a(5).
The subset {1,3,4} has 4 = 1 + 3, so is counted under a(4).
The subset {2,4,5} has 4 = 2 + 2, so is counted under a(5).
The a(0) = 0 through a(5) = 9 subsets:
  .  .  {1,2}  {1,2,3}  {2,4}      {1,2,5}
                        {1,2,4}    {1,4,5}
                        {1,3,4}    {2,3,5}
                        {2,3,4}    {2,4,5}
                        {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..88
• Steven R. Finch, Monoids of natural numbers, March 17, 2009.

Crossrefs

The complement w/o re-usable parts is A085489, first differences of A364755.

First differences of A093971.

The non-binary complement is A124506, first differences of A326083.

The complement is counted by A288728, first differences of A007865.

For partitions (not requiring n) we have A363225, strict A363226.

The case without re-usable parts is A364756, firsts differences of A088809.

The non-binary version is A365046, first differences of A364914.

A116861 and A364916 count linear combinations of strict partitions.

A364350 counts combination-free strict partitions, complement A364839.

A364913 counts combination-full partitions.

A365006 counts no positive combination-full strict ptns.

Cf. A050291, A051026, A151897, A236912, A237113, A237668, A326080, A364349, A364533, A364670.

Programs

Mathematica

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

Formula

First differences of A093971.

Extensions

a(21) onwards added (using A093971) by Andrew Howroyd, Jan 13 2024