A365070 - OEIS

%I #14 Jan 13 2024 16:47:15

%S 0,0,1,1,5,9,24,46,109,209,469,922,1932,3858,7952,15831,32214,64351,

%T 129813,259566,521681,1042703,2091626,4182470,8376007,16752524,

%U 33530042,67055129,134165194,268328011,536763582,1073523097,2147268041,4294505929,8589506814,17178978145

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

%C 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.

%H Andrew Howroyd, <a href="/A365070/b365070.txt">Table of n, a(n) for n = 0..88</a>

%H S. R. Finch, <a href="/A066062/a066062.pdf">Monoids of natural numbers</a>, March 17, 2009.

%F First differences of A093971.

%e The subset {1,3} has no element equal to the sum of two others, so is not counted under a(3).

%e The subset {3,4,5} has no element equal to the sum of two others, so is not counted under a(5).

%e The subset {1,3,4} has 4 = 1 + 3, so is counted under a(4).

%e The subset {2,4,5} has 4 = 2 + 2, so is counted under a(5).

%e The a(0) = 0 through a(5) = 9 subsets:

%e   .  .  {1,2}  {1,2,3}  {2,4}      {1,2,5}

%e                         {1,2,4}    {1,4,5}

%e                         {1,3,4}    {2,3,5}

%e                         {2,3,4}    {2,4,5}

%e                         {1,2,3,4}  {1,2,3,5}

%e                                    {1,2,4,5}

%e                                    {1,3,4,5}

%e                                    {2,3,4,5}

%e                                    {1,2,3,4,5}

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

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

%Y First differences of A093971.

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

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

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

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

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

%Y A116861 and A364916 count linear combinations of strict partitions.

%Y A364350 counts combination-free strict partitions, complement A364839.

%Y A364913 counts combination-full partitions.

%Y A365006 counts no positive combination-full strict ptns.

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

%K nonn

%O 0,5

%A _Gus Wiseman_, Aug 24 2023

%E a(21) onwards added (using A093971) by _Andrew Howroyd_, Jan 13 2024