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%I #27 Oct 23 2018 03:03:10
%S 1,1,2,3,27,165,676,3584,19108,80754,386776,1807342,8218582,114618650,
%T 1410831012,12144300991,126350575684
%N Number of cyclic arrangements of S={1,2,...,2n} such that the difference between any two neighbors is 3^k for some k=0,1,2,...
%C 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.
%C In this particular instance of NPC(n;S;P), all the terms with odd cycle lengths are necessarily zero.
%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 such cycles of length n=6 are:
%e C_1={1,2,3,6,5,4}, C_2={1,2,5,6,3,4}.
%e The first and last of the 27 such cycles of length n=10 are:
%e C_1={1,2,3,4,5,6,7,8,9,10}, C_27={1,4,7,8,5,2,3,6,9,10}.
%t A242520[n_] := Count[Map[lpf, Map[j1f, Permutations[Range[2, 2 n]]]], 0]/2;
%t j1f[x_] := Join[{1}, x, {1}];
%t lpf[x_] := Length[Select[Abs[Differences[x]], ! MemberQ[t, #] &]];
%t t = Table[3^k, {k, 0, 10}];
%t Join[{1}, Table[A242520[n], {n, 2, 5}]]
%t (* OR, a less simple, but more efficient implementation. *)
%t A242520[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 A242520[n, Join[perm, {new}],
%t Complement[Range[2, 2 n], perm, {new}]];
%t ];
%t Return[ct];
%t ];
%t ];
%t t = Table[3^k, {k, 0, 10}];
%t Join[{1}, Table[ct = 0; A242520[n, {1}, Range[2, 2 n]]/2, {n, 2, 8}]] (* _Robert Price_, Oct 22 2018 *)
%o (C++) See the link.
%Y Cf. A242519, A242521, A242522, A242523, A242524, A242525, A242526, A242527, A242528, A242529, A242530, A242531, A242532, A242533, A242534.
%K nonn,hard,more
%O 1,3
%A _Stanislav Sykora_, May 27 2014
%E a(14)-a(17) from _Andrew Howroyd_, Apr 05 2016
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