%I
%S 1,1,2,36,288,3888,200448,4257792,139511808,11813990400,532754620416
%N Number of cyclic arrangements of S={1,2,...,2n} such that the difference of any two neighbors is coprime to their sum.
%C a(n)=NPC(2n;S;P) is the count of all neighborproperty cycles for a specific set S of 2n elements and a specific pairproperty P. For more details, see the link and A242519.
%C Conjecture: in this case it seems that NPC(n;S;P)=0 for all odd n, so only the even ones are listed. This is definitely not the case when the property P is replaced by its negation (see A242534).
%H S. Sykora, <a href="http://dx.doi.org/10.3247/SL5Math14.002">On NeighborProperty Cycles</a>, <a href="http://ebyte.it/library/Library.html#math">Stan's Library</a>, Volume V, 2014.
%e For n=4, the only cycle is {1,2,3,4}.
%e The two solutions for n=6 are: C_1={1,2,3,4,5,6} and C_2={1,4,3,2,5,6}.
%t A242533[n_] := Count[Map[lpf, Map[j1f, Permutations[Range[2, 2 n]]]], 0]/2;
%t j1f[x_] := Join[{1}, x, {1}];
%t lpf[x_] := Length[Select[cpf[x], ! # &]];
%t cpf[x_] := Module[{i},
%t Table[CoprimeQ[x[[i]]  x[[i + 1]], x[[i]] + x[[i + 1]]], {i,
%t Length[x]  1}]];
%t Join[{1}, Table[A242533[n], {n, 2, 5}]]
%t (* OR, a less simple, but more efficient implementation. *)
%t A242533[n_, perm_, remain_] := Module[{opt, lr, i, new},
%t If[remain == {},
%t If[CoprimeQ[First[perm] + Last[perm], First[perm]  Last[perm]],
%t ct++];
%t Return[ct],
%t opt = remain; lr = Length[remain];
%t For[i = 1, i <= lr, i++,
%t new = First[opt]; opt = Rest[opt];
%t If[! CoprimeQ[Last[perm] + new, Last[perm]  new], Continue[]];
%t A242533[n, Join[perm, {new}],
%t Complement[Range[2, 2 n], perm, {new}]];
%t ];
%t Return[ct];
%t ];
%t ];
%t Join[{1}, Table[ct = 0; A242533[n, {1}, Range[2, 2 n]]/2, {n, 2, 6}] ](* _Robert Price_, Oct 25 2018 *)
%o (C++) See the link.
%Y Cf. A242519, A242520, A242521, A242522, A242523, A242524, A242525, A242526, A242527, A242528, A242529, A242530, A242531, A242532, A242534.
%K nonn,hard,more
%O 1,3
%A _Stanislav Sykora_, May 30 2014
%E a(10)a(11) from _Fausto A. C. Cariboni_, May 31 2017, Jun 01 2017
