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EXAMPLE
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There are four connected vertex sets of {{1,2},{1,3},{2,3}}, namely {1,2,3}, {1,2}, {1,3}, {2,3}; there are three connected vertex sets of {{1,2},{1,3}}, {{1,2},{2,3}}, and {{1,3},{2,3}} each; and there is one connected vertex set of {{1,2,3}}. So we have a total of a(3) = 4 + 3 * 3 + 1 = 14 connected vertex sets.
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MATHEMATICA
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nn=5;
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[OrderedQ[#], UnsameQ@@#, Length[Intersection@@s[[#]]]>0]&]}, If[c=={}, s, csm[Union[Append[Delete[s, List/@c[[1]]], multijoin@@s[[c[[1]]]]]]]]];
clutQ[eds_]:=And[UnsameQ@@eds, !Apply[Or, Outer[#1=!=#2&&Complement[#1, #2]=={}&, eds, eds, 1], {0, 1}], Length[csm[eds]]==1];
stableSets[u_, Q_]:=If[Length[u]==0, {{}}, With[{w=First[u]}, Join[stableSets[DeleteCases[u, w], Q], Prepend[#, w]&/@stableSets[DeleteCases[u, r_/; r==w||Q[r, w]||Q[w, r]], Q]]]];
swell[c_]:=Union@@FixedPointList[Union[ReplaceList[#1, {___, a:{___, x_, ___}, ___, b:{___, x_, ___}, ___}:>Union[a, b]]]&, c]
Table[Sum[Length[swell[c]], {c, Select[stableSets[Select[Subsets[Range[n]], Length[#]>1&], Complement[#1, #2]=={}&], And[Union@@#==Range[n], clutQ[#]]&]}], {n, nn}]
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