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Number of subsets of {1..n} containing all of their own first differences.
7

%I #12 Sep 06 2023 20:30:49

%S 1,2,4,6,10,14,23,34,58,96,171,302,565,1041,1969,3719,7105,13544,

%T 25999,49852,95949,184658,356129,687068,1327540,2566295,4966449,

%U 9617306,18640098,36150918,70166056,136272548,264844111,515036040,1002211421,1951345157,3801569113

%N Number of subsets of {1..n} containing all of their own first differences.

%H Rémy Sigrist, <a href="/A364671/a364671.txt">C++ program</a>

%e The subset {1,2,4,5,10,14} has differences (1,2,1,5,4) so is counted under a(14).

%e The a(0) = 1 through a(5) = 14 subsets:

%e {} {} {} {} {} {}

%e {1} {1} {1} {1} {1}

%e {2} {2} {2} {2}

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

%e {1,2} {4} {4}

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

%e {2,4} {1,2}

%e {1,2,3} {2,4}

%e {1,2,4} {1,2,3}

%e {1,2,3,4} {1,2,4}

%e {1,2,3,4}

%e {1,2,3,5}

%e {1,2,4,5}

%e {1,2,3,4,5}

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

%o (C++) See Links section.

%Y For differences of all strict pairs we have A054519, for partitions A007862.

%Y For "disjoint" instead of "subset" we have A364463, partitions A363260.

%Y For "non-disjoint" we have A364466, partitions A364467 (strict A364536).

%Y The complement is counted by A364672, partitions A364673, A364674, A364675.

%Y First differences of terms are A364752, complement A364753.

%Y Cf. A151897, A196723, A237667, A237668, A325325, A326083, A363225, A364345, A364464, A364537.

%K nonn

%O 0,2

%A _Gus Wiseman_, Aug 04 2023

%E More terms from _Rémy Sigrist_, Aug 06 2023