cp's OEIS Frontend

A242531 Number of cyclic arrangements of S={1,2,...,n} such that the difference of any two neighbors is a divisor of their sum.

%I A242531 #21 Jul 14 2020 20:30:17
%S A242531 0,1,1,1,1,4,3,9,26,82,46,397,283,1675,9938,19503,10247,97978,70478,
%T A242531 529383,3171795,7642285,3824927,48091810,116017829,448707198,
%U A242531 1709474581,6445720883,3009267707,51831264296
%N A242531 Number of cyclic arrangements of S={1,2,...,n} such that the difference of any two neighbors is a divisor of their sum.
%C A242531 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.
%H A242531 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 A242531 The only such cycle of length n=5 is {1,2,4,5,3}.
%e A242531 For n=7 there are three solutions: C_1={1,2,4,5,7,6,3}, C_2={1,2,4,6,7,5,3}, C_3={1,2,6,7,5,4,3}.
%t A242531 A242531[n_] := Count[Map[lpf, Map[j1f, Permutations[Range[2, n]]]], 0]/2;
%t A242531 j1f[x_] := Join[{1}, x, {1}];
%t A242531 dvf[x_] := Module[{i},
%t A242531    Table[Divisible[x[[i]] + x[[i + 1]], x[[i]] - x[[i + 1]]], {i,
%t A242531      Length[x] - 1}]];
%t A242531 lpf[x_] := Length[Select[dvf[x], ! # &]];
%t A242531 Join[{0, 1}, Table[A242531[n], {n, 3, 10}]]
%t A242531 (* OR, a less simple, but more efficient implementation. *)
%t A242531 A242531[n_, perm_, remain_] := Module[{opt, lr, i, new},
%t A242531    If[remain == {},
%t A242531      If[Divisible[First[perm] + Last[perm],
%t A242531        First[perm] - Last[perm]], ct++];
%t A242531      Return[ct],
%t A242531      opt = remain; lr = Length[remain];
%t A242531      For[i = 1, i <= lr, i++,
%t A242531       new = First[opt]; opt = Rest[opt];
%t A242531       If[! Divisible[Last[perm] + new, Last[perm] - new], Continue[]];
%t A242531       A242531[n, Join[perm, {new}],
%t A242531        Complement[Range[2, n], perm, {new}]];
%t A242531       ];
%t A242531      Return[ct];
%t A242531      ];
%t A242531    ];
%t A242531 Join[{0, 1}, Table[ct = 0; A242531[n, {1}, Range[2, n]]/2, {n, 3, 13}]] (* _Robert Price_, Oct 25 2018 *)
%o A242531 (C++) See the link.
%Y A242531 Cf. A242519, A242520, A242521, A242522, A242523, A242524, A242525, A242526, A242527, A242528, A242529, A242530, A242532, A242533, A242534.
%K A242531 nonn,hard,more
%O A242531 1,6
%A A242531 _Stanislav Sykora_, May 30 2014
%E A242531 a(24)-a(28) from _Fausto A. C. Cariboni_, May 25 2017
%E A242531 a(29) from _Fausto A. C. Cariboni_, Jul 09 2020
%E A242531 a(30) from _Fausto A. C. Cariboni_, Jul 14 2020