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A324512 Number of aperiodic n-gons.

%I A324512 #11 Aug 19 2019 16:42:58
%S A324512 1,0,0,0,10,42,357,2400,20142,180280,1814395,19944804,239500794,
%T A324512 3113326062,43589143560,653834280960,10461394943992,177843662409312,
%U A324512 3201186852863991,60822549182544440,1216451004087794832,25545471063559372750,562000363888803839989
%N A324512 Number of aperiodic n-gons.
%C A324512 We define an n-gon to be aperiodic if all n rotations of its vertex set act on the edge set to give distinct n-gons. These are different from aperiodic graphs and acyclic graphs but are similar to aperiodic sequences (A000740) and aperiodic arrays (A323867).
%H A324512 Andrew Howroyd, <a href="/A324512/b324512.txt">Table of n, a(n) for n = 1..200</a>
%H A324512 Gus Wiseman, <a href="/A324512/a324512.png">The a(5) = 10 aperiodic polygons</a>.
%H A324512 Gus Wiseman, <a href="/A324512/a324512_1.png">The a(6) = 42 aperiodic polygons</a>.
%F A324512 a(n) = n * A324513(n).
%e A324512 The a(5) = 10 aperiodic polygon edge sets:
%e A324512   {{1,2},{1,3},{2,4},{3,5},{4,5}}
%e A324512   {{1,2},{1,3},{2,5},{3,4},{4,5}}
%e A324512   {{1,2},{1,4},{2,3},{3,5},{4,5}}
%e A324512   {{1,2},{1,4},{2,5},{3,4},{3,5}}
%e A324512   {{1,2},{1,5},{2,4},{3,4},{3,5}}
%e A324512   {{1,3},{1,4},{2,3},{2,5},{4,5}}
%e A324512   {{1,3},{1,5},{2,3},{2,4},{4,5}}
%e A324512   {{1,3},{1,5},{2,4},{2,5},{3,4}}
%e A324512   {{1,4},{1,5},{2,3},{2,4},{3,5}}
%e A324512   {{1,4},{1,5},{2,3},{2,5},{3,4}}
%t A324512 rotgra[g_,m_]:=Sort[Sort/@(g/.k_Integer:>If[k==m,1,k+1])];
%t A324512 Table[Length[Select[Union[Sort[Sort/@Partition[#,2,1,1]]&/@Permutations[Range[n]]],UnsameQ@@Table[Nest[rotgra[#,n]&,#,j],{j,n}]&]],{n,8}]
%o A324512 (PARI) a(n)={if(n<3, n==1, (if(n%2, 0, -n*(n/2-1)!*2^(n/2-2)) + sumdiv(n, d, moebius(n/d)*eulerphi(n/d)*(n/d)^d*d!/n))/2)} \\ _Andrew Howroyd_, Aug 19 2019
%Y A324512 Cf. A000740, A008965, A027375, A059966, A060223, A192332, A275527, A323860, A323867, A323869, A324461, A324462, A324513, A324514.
%K A324512 nonn
%O A324512 1,5
%A A324512 _Gus Wiseman_, Mar 04 2019
%E A324512 Terms a(10) and beyond from _Andrew Howroyd_, Aug 19 2019