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 A242534 Number of cyclic arrangements of S={1,2,...,n} such that the difference of any two neighbors is not coprime to their sum. 16
 1, 0, 0, 0, 0, 0, 0, 0, 0, 72, 288, 3600, 17856, 174528, 2540160, 14768640, 101030400, 1458266112, 11316188160, 140951577600, 2659218508800, 30255151463424, 287496736542720, 5064092578713600, 76356431941939200, 987682437203558400, 19323690313219522560 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,10 COMMENTS a(n)=NPC(n;S;P) is the count of all neighbor-property cycles for a specific set S of n elements and a specific pair-property P. For more details, see the link and A242519. Compare this with A242533 where the property is inverted. LINKS Hiroaki Yamanouchi, Table of n, a(n) for n = 1..27 S. Sykora, On Neighbor-Property Cycles, Stan's Library, Volume V, 2014. EXAMPLE The first and the last of the 72 cycles for n=10 are: C_1={1,3,5,10,2,4,8,6,9,7} and C_72={1,7,5,10,8,4,2,6,3,9}. There are no solutions for cycle lengths from 2 to 9. MATHEMATICA A242534[n_] := Count[Map[lpf, Map[j1f, Permutations[Range[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[A242534[n], {n, 2, 10}]] (* OR, a less simple, but more efficient implementation. *) A242534[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[]];       A242534[n, Join[perm, {new}],        Complement[Range[2, n], perm, {new}]];       ];      Return[ct];      ];    ]; Join[{1}, Table[ct = 0; A242534[n, {1}, Range[2, n]]/2, {n, 2, 12}] ](* 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, A242533. Sequence in context: A279272 A173546 A308136 * A277430 A277991 A205627 Adjacent sequences:  A242531 A242532 A242533 * A242535 A242536 A242537 KEYWORD nonn,hard AUTHOR Stanislav Sykora, May 30 2014 EXTENSIONS a(19)-a(27) from Hiroaki Yamanouchi, Aug 30 2014 STATUS approved

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Last modified October 19 13:01 EDT 2019. Contains 328222 sequences. (Running on oeis4.)