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 A242533 Number of cyclic arrangements of S={1,2,...,2n} such that the difference of any two neighbors is coprime to their sum. 16
 1, 1, 2, 36, 288, 3888, 200448, 4257792, 139511808, 11813990400, 532754620416 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS a(n)=NPC(2n;S;P) is the count of all neighbor-property cycles for a specific set S of 2n elements and a specific pair-property P. For more details, see the link and A242519. 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). LINKS S. Sykora, On Neighbor-Property Cycles, Stan's Library, Volume V, 2014. EXAMPLE For n=4, the only cycle is {1,2,3,4}. The two solutions for n=6 are: C_1={1,2,3,4,5,6} and C_2={1,4,3,2,5,6}. MATHEMATICA A242533[n_] := Count[Map[lpf, Map[j1f, Permutations[Range[2, 2 n]]]], 0]/2; j1f[x_] := Join[{1}, x, {1}]; lpf[x_] := Length[Select[cpf[x], ! # &]]; cpf[x_] := Module[{i},    Table[CoprimeQ[x[[i]] - x[[i + 1]], x[[i]] + x[[i + 1]]], {i,      Length[x] - 1}]]; Join[{1}, Table[A242533[n], {n, 2, 5}]] (* OR, a less simple, but more efficient implementation. *) A242533[n_, perm_, remain_] := Module[{opt, lr, i, new},    If[remain == {},      If[CoprimeQ[First[perm] + Last[perm], First[perm] - Last[perm]],       ct++];      Return[ct],      opt = remain; lr = Length[remain];      For[i = 1, i <= lr, i++,       new = First[opt]; opt = Rest[opt];       If[! CoprimeQ[Last[perm] + new, Last[perm] - new], Continue[]];       A242533[n, Join[perm, {new}],        Complement[Range[2, 2 n], perm, {new}]];       ];      Return[ct];      ];    ]; Join[{1}, Table[ct = 0; A242533[n, {1}, Range[2, 2 n]]/2, {n, 2, 6}] ](* Robert Price, Oct 25 2018 *) PROG (C++) See the link. CROSSREFS Cf. A242519, A242520, A242521, A242522, A242523, A242524, A242525, A242526, A242527, A242528, A242529, A242530, A242531, A242532, A242534. Sequence in context: A341535 A074426 A082636 * A273325 A035603 A126735 Adjacent sequences:  A242530 A242531 A242532 * A242534 A242535 A242536 KEYWORD nonn,hard,more AUTHOR Stanislav Sykora, May 30 2014 EXTENSIONS a(10)-a(11) from Fausto A. C. Cariboni, May 31 2017, Jun 01 2017 STATUS approved

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Last modified September 27 04:10 EDT 2021. Contains 347673 sequences. (Running on oeis4.)