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A324512 Number of aperiodic n-gons. 4
1, 0, 0, 0, 10, 42, 357, 2400, 20142, 180280, 1814395, 19944804, 239500794, 3113326062, 43589143560, 653834280960, 10461394943992, 177843662409312, 3201186852863991, 60822549182544440, 1216451004087794832, 25545471063559372750, 562000363888803839989 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
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).
LINKS
FORMULA
a(n) = n * A324513(n).
EXAMPLE
The a(5) = 10 aperiodic polygon edge sets:
{{1,2},{1,3},{2,4},{3,5},{4,5}}
{{1,2},{1,3},{2,5},{3,4},{4,5}}
{{1,2},{1,4},{2,3},{3,5},{4,5}}
{{1,2},{1,4},{2,5},{3,4},{3,5}}
{{1,2},{1,5},{2,4},{3,4},{3,5}}
{{1,3},{1,4},{2,3},{2,5},{4,5}}
{{1,3},{1,5},{2,3},{2,4},{4,5}}
{{1,3},{1,5},{2,4},{2,5},{3,4}}
{{1,4},{1,5},{2,3},{2,4},{3,5}}
{{1,4},{1,5},{2,3},{2,5},{3,4}}
MATHEMATICA
rotgra[g_, m_]:=Sort[Sort/@(g/.k_Integer:>If[k==m, 1, k+1])];
Table[Length[Select[Union[Sort[Sort/@Partition[#, 2, 1, 1]]&/@Permutations[Range[n]]], UnsameQ@@Table[Nest[rotgra[#, n]&, #, j], {j, n}]&]], {n, 8}]
PROG
(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
CROSSREFS
Sequence in context: A321314 A348095 A027149 * A077541 A084036 A092117
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 04 2019
EXTENSIONS
Terms a(10) and beyond from Andrew Howroyd, Aug 19 2019
STATUS
approved

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Last modified June 25 13:13 EDT 2024. Contains 373705 sequences. (Running on oeis4.)