OFFSET
1,6
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.
LINKS
S. Sykora, On Neighbor-Property Cycles, Stan's Library, Volume V, 2014.
EXAMPLE
The only such cycle of length n=5 is {1,2,4,5,3}.
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}.
MATHEMATICA
A242531[n_] := Count[Map[lpf, Map[j1f, Permutations[Range[2, n]]]], 0]/2;
j1f[x_] := Join[{1}, x, {1}];
dvf[x_] := Module[{i},
Table[Divisible[x[[i]] + x[[i + 1]], x[[i]] - x[[i + 1]]], {i,
Length[x] - 1}]];
lpf[x_] := Length[Select[dvf[x], ! # &]];
Join[{0, 1}, Table[A242531[n], {n, 3, 10}]]
(* OR, a less simple, but more efficient implementation. *)
A242531[n_, perm_, remain_] := Module[{opt, lr, i, new},
If[remain == {},
If[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[! Divisible[Last[perm] + new, Last[perm] - new], Continue[]];
A242531[n, Join[perm, {new}],
Complement[Range[2, n], perm, {new}]];
];
Return[ct];
];
];
Join[{0, 1}, Table[ct = 0; A242531[n, {1}, Range[2, n]]/2, {n, 3, 13}]] (* Robert Price, Oct 25 2018 *)
PROG
(C++) See the link.
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Stanislav Sykora, May 30 2014
EXTENSIONS
a(24)-a(28) from Fausto A. C. Cariboni, May 25 2017
a(29) from Fausto A. C. Cariboni, Jul 09 2020
a(30) from Fausto A. C. Cariboni, Jul 14 2020
STATUS
approved