%I #10 Aug 07 2023 11:32:47

%S 0,0,0,2,4,12,23,53,104,218,437,893,1785,3620,7264,14634,29382,59097,

%T 118617,238291,478191,959867,1925681,3863365,7748136,15538461,

%U 31154278,62458007,125194936,250924636,502855774,1007635332,2018912085,4044775367,8102759211,16230735448,32509514412,65110826347

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

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

%e The a(1) = 0 through a(6) = 23 subsets:

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

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

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

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

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

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

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

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

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

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

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

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

%e {4,5,6}

%e {1,2,5,6}

%e {1,3,4,6}

%e {1,3,5,6}

%e {1,4,5,6}

%e {2,3,4,6}

%e {2,3,5,6}

%e {2,4,5,6}

%e {3,4,5,6}

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

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

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

%Y Partial sums are A364672, complement A364671.

%Y The complement is counted by A364752.

%Y A054519 counts subsets containing differences, A326083 containing sums.

%Y A364463 counts subsets disjoint from differences, complement A364466.

%Y A364673, A364674, A364675 count partitions containing differences.

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

%K nonn

%O 0,4

%A _Gus Wiseman_, Aug 06 2023

%E More terms from _Giorgos Kalogeropoulos_, Aug 07 2023