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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.
16
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, 44370, 0, 0, 0, 0, 2032951, 8840796, 12791922, 101519154, 0, 0
OFFSET
1,14
COMMENTS
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.
For this property P and sets {0,1,2,...,n-1} or {1,2,...,n} the problem does not appear to have any solution.
a(40)=a(41)=a(42)=a(43)=a(46)=a(47)=0. - Fausto A. C. Cariboni, May 17 2017
EXAMPLE
The shortest such cycle is of length n=9: {2,4,8,10,5,7,9,3,6}.
The next a(n)>0 occurs for n=14 and has 20 solutions.
The first and the last of these are:
C_1={2,4,8,10,5,7,14,12,15,13,11,9,3,6},
C_2={2,4,12,15,13,11,9,3,5,7,14,10,8,6}.
MATHEMATICA
A242532[n_] := Count[Map[lpf, Map[j2f, Permutations[Range[3, n + 1]]]], 0]/2;
j2f[x_] := Join[{2}, x, {2}];
dvf[x_] := Module[{i},
Table[Abs[x[[i]] - x[[i + 1]]] > 1 &&
Divisible[x[[i]] + x[[i + 1]], x[[i]] - x[[i + 1]]], {i,
Length[x] - 1}]];
lpf[x_] := Length[Select[dvf[x], ! # &]];
Table[A242532[n], {n, 1, 10}]
(* OR, a less simple, but more efficient implementation. *)
A242532[n_, perm_, remain_] := Module[{opt, lr, i, new},
If[remain == {},
If[Abs[First[perm] - Last[perm]] > 1 &&
Divisible[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[Abs[Last[perm] - new] <= 1 || !
Divisible[Last[perm] + new, Last[perm] - new], Continue[]];
A242532[n, Join[perm, {new}],
Complement[Range[3, n + 1], perm, {new}]];
];
Return[ct];
];
];
Table[ct = 0; A242532[n, {2}, Range[3, n + 1]]/2, {n, 1, 15}] (* Robert Price, Oct 25 2018 *)
PROG
(C++) See the link.
KEYWORD
nonn,hard,more
AUTHOR
Stanislav Sykora, May 30 2014
EXTENSIONS
a(29)-a(37) from Fausto A. C. Cariboni, May 17 2017
STATUS
approved