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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

%I #26 Oct 24 2018 22:27:50

%S 0,0,0,0,0,0,0,0,0,0,0,0,2,4,6,9,42,231,1052,3818,10086,27892,90076,

%T 310301,993680,4663558,22038882,162588454,1246422151,8655752023,

%U 58951670318,347675502245

%N 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 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 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 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[n_] := Count[Map[lpf, Map[j1f, Permutations[Range[2, n]]]], 0]/2;

%t j1f[x_] := Join[{1}, x, {1}];

%t lpf[x_] := Length[Select[Abs[Differences[x]], ! MemberQ[t, #] &]];

%t t = Flatten[Table[b^k, {k, 2, 5}, {b, 2, 5}]];

%t Table[A242521[n], {n, 1, 10}]

%t (* OR, a less simple, but more efficient implementation. *)

%t A242521[n_, perm_, remain_] := Module[{opt, lr, i, new},

%t If[remain == {},

%t If[MemberQ[t, Abs[First[perm] - Last[perm]]], 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[! MemberQ[t, Abs[Last[perm] - new]], Continue[]];

%t A242521[n, Join[perm, {new}],

%t Complement[Range[2, n], perm, {new}]];

%t ];

%t Return[ct];

%t ];

%t ];

%t t = Flatten[Table[b^k, {k, 2, 5}, {b, 2, 5}]];

%t Table[ct = 0; A242521[n, {1}, Range[2, n]]/2, {n, 1, 18}] (* _Robert Price_, Oct 24 2018 *)

%o (C++) See the link.

%Y Cf. A242519, A242520, A242522, A242523, A242524, A242525, A242526, A242527, A242528, A242529, A242530, A242531, A242532, A242533, A242534.

%K nonn,hard,more

%O 1,13

%A _Stanislav Sykora_, May 27 2014

%E a(27)-a(30) from _Max Alekseyev_, Jul 12 2014

%E a(31)-a(32) from _Fausto A. C. Cariboni_, May 17 2017, May 24 2017