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 A242521 Number of cyclic arrangements (up to direction) of {1,2,...,n} such that the difference between any two neighbors is b^k for some b>1 and k>1. 16
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 6, 9, 42, 231, 1052, 3818, 10086, 27892, 90076, 310301, 993680, 4663558, 22038882, 162588454, 1246422151, 8655752023, 58951670318, 347675502245 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,13 COMMENTS a(n)=NPC(n;S;P) is the count of all neighbor-property cycles for a specific set S={1,2,...,n} of n elements and a specific pair-property P. For more details, see the link and A242519. LINKS S. Sykora, On Neighbor-Property Cycles, Stan's Library, Volume V, 2014. EXAMPLE The two cycles of length n=13 (the smallest n such that a(n)>0) are: C_1={1,5,9,13,4,8,12,3,7,11,2,6,10}, C_2={1,9,5,13,4,8,12,3,7,11,2,6,10}. MATHEMATICA A242521[n_] := Count[Map[lpf, Map[j1f, Permutations[Range[2, n]]]], 0]/2; j1f[x_] := Join[{1}, x, {1}]; lpf[x_] := Length[Select[Abs[Differences[x]], ! MemberQ[t, #] &]]; t = Flatten[Table[b^k, {k, 2, 5}, {b, 2, 5}]]; Table[A242521[n], {n, 1, 10}] (* OR, a less simple, but more efficient implementation. *) A242521[n_, perm_, remain_] := Module[{opt, lr, i, new},    If[remain == {},      If[MemberQ[t, Abs[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[! MemberQ[t, Abs[Last[perm] - new]], Continue[]];       A242521[n, Join[perm, {new}],        Complement[Range[2, n], perm, {new}]];       ];      Return[ct];      ];    ]; t = Flatten[Table[b^k, {k, 2, 5}, {b, 2, 5}]]; Table[ct = 0; A242521[n, {1}, Range[2, n]]/2, {n, 1, 18}] (* Robert Price, Oct 24 2018 *) PROG (C++) See the link. CROSSREFS Cf. A242519, A242520, A242522, A242523, A242524, A242525, A242526, A242527, A242528, A242529, A242530, A242531, A242532, A242533, A242534. Sequence in context: A271927 A227526 A093081 * A073659 A073661 A079052 Adjacent sequences:  A242518 A242519 A242520 * A242522 A242523 A242524 KEYWORD nonn,hard,more AUTHOR Stanislav Sykora, May 27 2014 EXTENSIONS a(27)-a(30) from Max Alekseyev, Jul 12 2014 a(31)-a(32) from Fausto A. C. Cariboni, May 17 2017, May 24 2017 STATUS approved

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Last modified November 12 12:43 EST 2018. Contains 317109 sequences. (Running on oeis4.)