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

0,4

Comments

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

Links

Table of n, a(n) for n=0..37.

Example

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}

Mathematica

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

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.

Sequence in context: A303030 A291404 A295954 * A052416 A059322 A163908

Adjacent sequences: A364750 A364751 A364752 * A364754 A364755 A364756

Keyword

nonn

Author

Gus Wiseman, Aug 06 2023

Extensions

More terms from Giorgos Kalogeropoulos, Aug 07 2023

Status

approved