A288728 Number of sum-free sets that can be created by adding n to all sum-free sets [1..n-1].

1, 1, 3, 3, 7, 8, 18, 19, 47, 43, 102, 116, 238, 240, 553, 554, 1185, 1259, 2578, 2607, 5873, 5526, 11834, 12601, 24692, 24390, 53735, 52534, 107445, 107330, 218727, 215607, 461367, 427778, 891039, 910294, 1804606, 1706828, 3695418, 3411513, 7136850, 6892950

Offset

1,3

Comments

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

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

Links

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..88

Eric Weisstein's World of Mathematics, Sum-Free Set

Formula

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

Example

1 can be added to {};
2 can be added to {} but not {1};
3 can be added to {},{1},{2};
4 can be added to {},{1},{3} but not {2},{1,3},{2,3}.
From Gus Wiseman, Aug 12 2023: (Start)
The a(1) = 1 through a(7) = 18 sum-free sets with maximum n:
  {1}  {2}  {3}    {4}    {5}      {6}      {7}
            {1,3}  {1,4}  {1,5}    {1,6}    {1,7}
            {2,3}  {3,4}  {2,5}    {2,6}    {2,7}
                          {3,5}    {4,6}    {3,7}
                          {4,5}    {5,6}    {4,7}
                          {1,3,5}  {1,4,6}  {5,7}
                          {3,4,5}  {2,5,6}  {6,7}
                                   {4,5,6}  {1,3,7}
                                            {1,4,7}
                                            {1,5,7}
                                            {2,3,7}
                                            {2,6,7}
                                            {3,5,7}
                                            {4,5,7}
                                            {4,6,7}
                                            {5,6,7}
                                            {1,3,5,7}
                                            {4,5,6,7}
(End)

Mathematica

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

Crossrefs

Cf. A007865.

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

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

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

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

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

Sequence in context: A286110 A116157 A056357 * A213942 A144554 A177936

Adjacent sequences: A288725 A288726 A288727 * A288729 A288730 A288731

Keyword

nonn,changed

Author

Ben Burns, Jun 14 2017

Status

approved