A365071 - OEIS

%I #14 Dec 13 2024 09:37:29

%S 0,1,2,3,6,9,15,23,40,55,94,132,210,298,476,644,1038,1406,2149,2965,

%T 4584,6077,9426,12648,19067,25739,38958,51514,78459,104265,155436,

%U 208329,312791,411886,620780,823785,1224414,1631815,2437015,3217077,4822991

%N Number of subsets of {1..n} containing n such that no element is a sum of distinct other elements. A variation of non-binary sum-free subsets without re-usable elements.

%C The complement is counted by A365069. The binary version is A364755, complement A364756. For re-usable parts we have A288728, complement A365070.

%H Andrew Howroyd, <a href="/A365071/b365071.txt">Table of n, a(n) for n = 0..85</a>

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

%F a(n) + A365069(n) = 2^(n-1).

%F First differences of A151897.

%e The subset {1,3,4,6} has 4 = 1 + 3 so is not counted under a(6).

%e The subset {2,3,4,5,6} has 6 = 2 + 4 and 4 = 1 + 3 so is not counted under a(6).

%e The a(0) = 0 through a(6) = 15 subsets:

%e   .  {1}  {2}    {3}    {4}      {5}      {6}

%e           {1,2}  {1,3}  {1,4}    {1,5}    {1,6}

%e                  {2,3}  {2,4}    {2,5}    {2,6}

%e                         {3,4}    {3,5}    {3,6}

%e                         {1,2,4}  {4,5}    {4,6}

%e                         {2,3,4}  {1,2,5}  {5,6}

%e                                  {1,3,5}  {1,2,6}

%e                                  {2,4,5}  {1,3,6}

%e                                  {3,4,5}  {1,4,6}

%e                                           {2,3,6}

%e                                           {2,5,6}

%e                                           {3,4,6}

%e                                           {3,5,6}

%e                                           {4,5,6}

%e                                           {3,4,5,6}

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

%Y First differences of A151897.

%Y The version with re-usable parts is A288728 first differences of A007865.

%Y The binary version is A364755, first differences of A085489.

%Y The binary complement is A364756, first differences of A088809.

%Y The complement is counted by A365069, first differences of A364534.

%Y The complement w/ re-usable parts is A365070, first differences of A093971.

%Y A108917 counts knapsack partitions, strict A275972.

%Y A124506 counts combination-free subsets, differences of A326083.

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

%Y A365046 counts combination-full subsets, differences of A364914.

%Y Partitions: A236912, A237113, A237668, A364532, A364272, A364349, A364913.

%Y Cf. A050291, A095944, A103580, A324741, A326080, A326117, A341507, A364533.

%K nonn

%O 0,3

%A _Gus Wiseman_, Aug 26 2023

%E a(14) onwards added (using A151897) by _Andrew Howroyd_, Jan 13 2024