A364753 Number of subsets of {1..n} containing n but not containing all first differences.

0, 0, 0, 2, 4, 12, 23, 53, 104, 218, 437, 893, 1785, 3620, 7264, 14634, 29382, 59097, 118617, 238291, 478191, 959867, 1925681, 3863365, 7748136, 15538461, 31154278, 62458007, 125194936, 250924636, 502855774, 1007635332, 2018912085, 4044775367, 8102759211, 16230735448, 32509514412, 65110826347

Offset: 0

Author

Gus Wiseman, Aug 06 2023

Keywords

• nonn

Comments

In other words, subsets containing both n and some element that is not the difference of two consecutive elements.

Examples

			The a(1) = 0 through a(6) = 23 subsets:
  .  .  {1,3}  {1,4}    {1,5}      {1,6}
        {2,3}  {3,4}    {2,5}      {2,6}
               {1,3,4}  {3,5}      {4,6}
               {2,3,4}  {4,5}      {5,6}
                        {1,2,5}    {1,2,6}
                        {1,3,5}    {1,3,6}
                        {1,4,5}    {1,4,6}
                        {2,3,5}    {1,5,6}
                        {2,4,5}    {2,3,6}
                        {3,4,5}    {2,5,6}
                        {1,3,4,5}  {3,4,6}
                        {2,3,4,5}  {3,5,6}
                                   {4,5,6}
                                   {1,2,5,6}
                                   {1,3,4,6}
                                   {1,3,5,6}
                                   {1,4,5,6}
                                   {2,3,4,6}
                                   {2,3,5,6}
                                   {2,4,5,6}
                                   {3,4,5,6}
                                   {1,3,4,5,6}
                                   {2,3,4,5,6}
		

Crossrefs

Partial sums are A364672, complement A364671.

The complement is counted by A364752.

A054519 counts subsets containing differences, A326083 containing sums.

A364463 counts subsets disjoint from differences, complement A364466.

A364673, A364674, A364675 count partitions containing differences.

Cf. A151897, A196723, A237668, A325325, A364345, A364464, A364537.

Programs

Mathematica

Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&!SubsetQ[#,Differences[#]]&]],{n,0,10}]

Extensions

More terms from Giorgos Kalogeropoulos, Aug 07 2023