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

0,5

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.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..88

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

Formula

First differences of A093971.

Example

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}

Mathematica

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

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.

Sequence in context: A272256 A132354 A146419 * A162907 A165345 A177240

Adjacent sequences: A365067 A365068 A365069 * A365071 A365072 A365073

Keyword

nonn

Author

Gus Wiseman, Aug 24 2023

Extensions

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

Status

approved