A288728 - OEIS

%I #22 Aug 12 2023 23:02:04

%S 1,1,3,3,7,8,18,19,47,43,102,116,238,240,553,554,1185,1259,2578,2607,

%T 5873,5526,11834,12601,24692,24390,53735,52534,107445,107330,218727,

%U 215607,461367,427778,891039,910294,1804606,1706828,3695418,3411513,7136850,6892950

%N Number of sum-free sets that can be created by adding n to all sum-free sets [1..n-1].

%C Using the standard definition of sum-free set, this is simply the difference of successive terms in A007865.

%C Number of subsets of {1..n} containing n but not containing the sum of any other two elements (repeats allowed). Also the number of sum-free sets (A007865) with maximum n. - _Gus Wiseman_, Aug 12 2023

%H Fausto A. C. Cariboni, <a href="/A288728/b288728.txt">Table of n, a(n) for n = 1..88</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Sum-FreeSet.html">Sum-Free Set</a>

%F a(n) = A007865(n) - A007865(n-1).

%e 1 can be added to {};

%e 2 can be added to {} but not {1};

%e 3 can be added to {},{1},{2};

%e 4 can be added to {},{1},{3} but not {2},{1,3},{2,3}.

%e From _Gus Wiseman_, Aug 12 2023: (Start)

%e The a(1) = 1 through a(7) = 18 sum-free sets with maximum n:

%e   {1}  {2}  {3}    {4}    {5}      {6}      {7}

%e             {1,3}  {1,4}  {1,5}    {1,6}    {1,7}

%e             {2,3}  {3,4}  {2,5}    {2,6}    {2,7}

%e                           {3,5}    {4,6}    {3,7}

%e                           {4,5}    {5,6}    {4,7}

%e                           {1,3,5}  {1,4,6}  {5,7}

%e                           {3,4,5}  {2,5,6}  {6,7}

%e                                    {4,5,6}  {1,3,7}

%e                                             {1,4,7}

%e                                             {1,5,7}

%e                                             {2,3,7}

%e                                             {2,6,7}

%e                                             {3,5,7}

%e                                             {4,5,7}

%e                                             {4,6,7}

%e                                             {5,6,7}

%e                                             {1,3,5,7}

%e                                             {4,5,6,7}

%e (End)

%t Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&Intersection[#,Total/@Tuples[#,2]]=={}&]],{n,10}] (* _Gus Wiseman_, Aug 12 2023 *)

%Y Cf. A007865.

%Y For non-binary sum-free subsets of {1..n} we have A237667.

%Y For sum-free partitions we have A364345, without re-using parts A236912.

%Y Without re-using parts we have A364755, diffs of A085489 (non-bin A151897).

%Y The complement without re-using parts is A364756, differences of A088809.

%Y Cf. A002865, A093971, A237113, A237668, A326083, A364347, A364348, A364534.

%K nonn

%O 1,3

%A _Ben Burns_, Jun 14 2017