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 A192332 For n >= 3, draw a regular n-sided polygon and its n(n-3)/2 diagonals, so there are n(n-1)/2 lines; a(n) is the number of ways to choose a subset of these lines (subsets differing by a rotation are regarded as identical). a(1)=1, a(2)=2 by convention. 16
 1, 2, 4, 22, 208, 5560, 299600, 33562696, 7635498336, 3518440564544, 3275345183542208, 6148914696963883712, 23248573454127484129024, 176848577040808821410837120, 2704321280486889389864215362560, 83076749736557243209409446411255936, 5124252113632955685095523500148980125696, 634332307869315502692705867068871886072665600 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Suggested by A192314. Also the number of graphical necklaces with n vertices. We define a graphical necklace to be a simple graph that is minimal among all n rotations of the vertices. Alternatively, it is an equivalence class of simple graphs under rotation of the vertices. These are a kind of partially labeled graphs. - Gus Wiseman, Mar 04 2019 LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..80 FORMULA a(n) = (1/n)*(Sum_{d|n, d odd} phi(d)*2^(n*(n-1)/(2*d)) + Sum_{d|n, d even} phi(d)*2^(n^2/(2*d))). EXAMPLE From Gus Wiseman, Mar 04 2019: (Start) Inequivalent representatives of the a(1) = 1 through a(4) = 22 graphical necklace edge-sets:   {}  {}      {}              {}       {{12}}  {{12}}          {{12}}               {{12}{13}}      {{13}}               {{12}{13}{23}}  {{12}{13}}                               {{12}{14}}                               {{12}{24}}                               {{12}{34}}                               {{13}{24}}                               {{12}{13}{14}}                               {{12}{13}{23}}                               {{12}{13}{24}}                               {{12}{13}{34}}                               {{12}{14}{23}}                               {{12}{24}{34}}                               {{12}{13}{14}{23}}                               {{12}{13}{14}{24}}                               {{12}{13}{14}{34}}                               {{12}{13}{24}{34}}                               {{12}{14}{23}{34}}                               {{12}{13}{14}{23}{24}}                               {{12}{13}{14}{23}{34}}                               {{12}{13}{14}{23}{24}{34}} (End) MAPLE with(numtheory); f:=proc(n) local t0, t1, d; t0:=0; t1:=divisors(n); for d in t1 do if d mod 2 = 0 then t0:=t0+phi(d)*2^(n^2/(2*d)) else t0:=t0+phi(d)*2^(n*(n-1)/(2*d)); fi; od; t0/n; end; [seq(f(n), n=1..30)]; MATHEMATICA Table[ 1/n* Plus @@ Map[Function[d, EulerPhi[d]*2^((n*(n - Mod[d, 2])/2)/d)], Divisors[n]], {n, 1, 20}]  (* Olivier Gérard, Aug 27 2011 *) rotgra[g_, m_]:=Sort[Sort/@(g/.k_Integer:>If[k==m, 1, k+1])]; Table[Length[Select[Subsets[Subsets[Range[n], {2}]], #=={}||#==First[Sort[Table[Nest[rotgra[#, n]&, #, j], {j, n}]]]&]], {n, 0, 5}] (* Gus Wiseman, Mar 04 2019 *) PROG (PARI) a(n) = sumdiv(n, d, if (d%2, eulerphi(d)*2^(n*(n-1)/(2*d)), eulerphi(d)*2^(n^2/(2*d))))/n; \\ Michel Marcus, Mar 08 2019 CROSSREFS Cf. A192314, A191563 (orbits under dihedral group). Cf. A000031, A000939 (cycle necklaces), A008965, A059966, A060223, A061417, A086675 (digraph version), A184271, A275527, A323858, A324461, A324463, A324464. Sequence in context: A283322 A019025 A264729 * A324603 A322520 A018279 Adjacent sequences:  A192329 A192330 A192331 * A192333 A192334 A192335 KEYWORD nonn AUTHOR N. J. A. Sloane, Jun 28 2011 STATUS approved

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Last modified February 25 02:54 EST 2020. Contains 332217 sequences. (Running on oeis4.)