login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation to keep the OEIS running. In 2018 we replaced the server with a faster one, added 20000 new sequences, and reached 7000 citations (often saying "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A242533 Number of cyclic arrangements of S={1,2,...,2n} such that the difference of any two neighbors is coprime to their sum. 16
1, 1, 2, 36, 288, 3888, 200448, 4257792, 139511808, 11813990400, 532754620416 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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.

Conjecture: in this case it seems that NPC(n;S;P)=0 for all odd n, so only the even ones are listed. This is definitely not the case when the property P is replaced by its negation (see A242534).

LINKS

Table of n, a(n) for n=1..11.

S. Sykora, On Neighbor-Property Cycles, Stan's Library, Volume V, 2014.

EXAMPLE

For n=4, the only cycle is {1,2,3,4}.

The two solutions for n=6 are: C_1={1,2,3,4,5,6} and C_2={1,4,3,2,5,6}.

MATHEMATICA

A242533[n_] := Count[Map[lpf, Map[j1f, Permutations[Range[2, 2 n]]]], 0]/2;

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

lpf[x_] := Length[Select[cpf[x], ! # &]];

cpf[x_] := Module[{i},

   Table[CoprimeQ[x[[i]] - x[[i + 1]], x[[i]] + x[[i + 1]]], {i,

     Length[x] - 1}]];

Join[{1}, Table[A242533[n], {n, 2, 5}]]

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

A242533[n_, perm_, remain_] := Module[{opt, lr, i, new},

   If[remain == {},

     If[CoprimeQ[First[perm] + Last[perm], First[perm] - Last[perm]],

      ct++];

     Return[ct],

     opt = remain; lr = Length[remain];

     For[i = 1, i <= lr, i++,

      new = First[opt]; opt = Rest[opt];

      If[! CoprimeQ[Last[perm] + new, Last[perm] - new], Continue[]];

      A242533[n, Join[perm, {new}],

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

      ];

     Return[ct];

     ];

   ];

Join[{1}, Table[ct = 0; A242533[n, {1}, Range[2, 2 n]]/2, {n, 2, 6}] ](* Robert Price, Oct 25 2018 *)

PROG

(C++) See the link.

CROSSREFS

Cf. A242519, A242520, A242521, A242522, A242523, A242524, A242525, A242526, A242527, A242528, A242529, A242530, A242531, A242532, A242534.

Sequence in context: A099903 A074426 A082636 * A273325 A035603 A126735

Adjacent sequences:  A242530 A242531 A242532 * A242534 A242535 A242536

KEYWORD

nonn,hard,more

AUTHOR

Stanislav Sykora, May 30 2014

EXTENSIONS

a(10)-a(11) from Fausto A. C. Cariboni, May 31 2017, Jun 01 2017

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 16 07:12 EST 2018. Contains 318158 sequences. (Running on oeis4.)