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EXAMPLE
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The a(3) = 8 edge-sets:
{12,13,23} {11,12,13,23} {11,12,13,22,23} {11,12,13,22,23,33}
{12,13,22,23} {11,12,13,23,33}
{12,13,23,33} {12,13,22,23,33}
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MATHEMATICA
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Table[Length[Select[Subsets[Select[Tuples[Range[n], 2], OrderedQ]], FindHamiltonianCycle[Graph[Range[n], #]]!={}&]], {n, 0, 5}]
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