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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.

%I A242521 #26 Oct 24 2018 22:27:50
%S A242521 0,0,0,0,0,0,0,0,0,0,0,0,2,4,6,9,42,231,1052,3818,10086,27892,90076,
%T A242521 310301,993680,4663558,22038882,162588454,1246422151,8655752023,
%U A242521 58951670318,347675502245
%N 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.
%C A242521 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.
%H A242521 S. Sykora, <a href="http://dx.doi.org/10.3247/SL5Math14.002">On Neighbor-Property Cycles</a>, <a href="http://ebyte.it/library/Library.html#math">Stan's Library</a>, Volume V, 2014.
%e A242521 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}.
%t A242521 A242521[n_] := Count[Map[lpf, Map[j1f, Permutations[Range[2, n]]]], 0]/2;
%t A242521 j1f[x_] := Join[{1}, x, {1}];
%t A242521 lpf[x_] := Length[Select[Abs[Differences[x]], ! MemberQ[t, #] &]];
%t A242521 t = Flatten[Table[b^k, {k, 2, 5}, {b, 2, 5}]];
%t A242521 Table[A242521[n], {n, 1, 10}]
%t A242521 (* OR, a less simple, but more efficient implementation. *)
%t A242521 A242521[n_, perm_, remain_] := Module[{opt, lr, i, new},
%t A242521    If[remain == {},
%t A242521      If[MemberQ[t, Abs[First[perm] - Last[perm]]], ct++];
%t A242521      Return[ct],
%t A242521      opt = remain; lr = Length[remain];
%t A242521      For[i = 1, i <= lr, i++,
%t A242521       new = First[opt]; opt = Rest[opt];
%t A242521       If[! MemberQ[t, Abs[Last[perm] - new]], Continue[]];
%t A242521       A242521[n, Join[perm, {new}],
%t A242521        Complement[Range[2, n], perm, {new}]];
%t A242521       ];
%t A242521      Return[ct];
%t A242521      ];
%t A242521    ];
%t A242521 t = Flatten[Table[b^k, {k, 2, 5}, {b, 2, 5}]];
%t A242521 Table[ct = 0; A242521[n, {1}, Range[2, n]]/2, {n, 1, 18}] (* _Robert Price_, Oct 24 2018 *)
%o A242521 (C++) See the link.
%Y A242521 Cf. A242519, A242520, A242522, A242523, A242524, A242525, A242526, A242527, A242528, A242529, A242530, A242531, A242532, A242533, A242534.
%K A242521 nonn,hard,more
%O A242521 1,13
%A A242521 _Stanislav Sykora_, May 27 2014
%E A242521 a(27)-a(30) from _Max Alekseyev_, Jul 12 2014
%E A242521 a(31)-a(32) from _Fausto A. C. Cariboni_, May 17 2017, May 24 2017