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A242532 Number of cyclic arrangements of S={2,3,...,n+1} such that the difference of any two neighbors is greater than 1, and a divisor of their sum.

%I A242532 #20 Oct 25 2018 21:25:43
%S A242532 0,0,0,0,0,0,0,0,1,0,0,0,0,20,39,0,0,0,0,319,967,0,0,1464,6114,16856,
%T A242532 44370,0,0,0,0,2032951,8840796,12791922,101519154,0,0
%N A242532 Number of cyclic arrangements of S={2,3,...,n+1} such that the difference of any two neighbors is greater than 1, and a divisor of their sum.
%C A242532 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.
%C A242532 For this property P and sets {0,1,2,...,n-1} or {1,2,...,n} the problem does not appear to have any solution.
%C A242532 a(40)=a(41)=a(42)=a(43)=a(46)=a(47)=0. - _Fausto A. C. Cariboni_, May 17 2017
%H A242532 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 A242532 The shortest such cycle is of length n=9: {2,4,8,10,5,7,9,3,6}.
%e A242532 The next a(n)>0 occurs for n=14 and has 20 solutions.
%e A242532 The first and the last of these are:
%e A242532 C_1={2,4,8,10,5,7,14,12,15,13,11,9,3,6},
%e A242532 C_2={2,4,12,15,13,11,9,3,5,7,14,10,8,6}.
%t A242532 A242532[n_] := Count[Map[lpf, Map[j2f, Permutations[Range[3, n + 1]]]], 0]/2;
%t A242532 j2f[x_] := Join[{2}, x, {2}];
%t A242532 dvf[x_] := Module[{i},
%t A242532    Table[Abs[x[[i]] - x[[i + 1]]] > 1 &&
%t A242532      Divisible[x[[i]] + x[[i + 1]], x[[i]] - x[[i + 1]]], {i,
%t A242532      Length[x] - 1}]];
%t A242532 lpf[x_] := Length[Select[dvf[x], ! # &]];
%t A242532 Table[A242532[n], {n, 1, 10}]
%t A242532 (* OR, a less simple, but more efficient implementation. *)
%t A242532 A242532[n_, perm_, remain_] := Module[{opt, lr, i, new},
%t A242532    If[remain == {},
%t A242532      If[Abs[First[perm] - Last[perm]] > 1 &&
%t A242532        Divisible[First[perm] + Last[perm], First[perm] - Last[perm]],
%t A242532       ct++];
%t A242532      Return[ct],
%t A242532      opt = remain; lr = Length[remain];
%t A242532      For[i = 1, i <= lr, i++,
%t A242532       new = First[opt]; opt = Rest[opt];
%t A242532       If[Abs[Last[perm] - new] <= 1 || !
%t A242532          Divisible[Last[perm] + new, Last[perm] - new], Continue[]];
%t A242532       A242532[n, Join[perm, {new}],
%t A242532        Complement[Range[3, n + 1], perm, {new}]];
%t A242532       ];
%t A242532      Return[ct];
%t A242532      ];
%t A242532    ];
%t A242532 Table[ct = 0; A242532[n, {2}, Range[3, n + 1]]/2, {n, 1, 15}] (* _Robert Price_, Oct 25 2018 *)
%o A242532 (C++) See the link.
%Y A242532 Cf. A242519, A242520, A242521, A242522, A242523, A242524, A242525, A242526, A242527, A242528, A242529, A242530, A242531, A242533, A242534.
%K A242532 nonn,hard,more
%O A242532 1,14
%A A242532 _Stanislav Sykora_, May 30 2014
%E A242532 a(29)-a(37) from _Fausto A. C. Cariboni_, May 17 2017