OFFSET
1,5
COMMENTS
Here, a(n) = NPC(2n;S;P) is the count of all neighbor-property cycles for a specific set S of 2n elements and a pair-property P. For more details, see the link and A242519.
In this case the property P is the Gray condition. The choice of the set S is important; when it is replaced by {0,1,2,...,2n-1}, the sequence changes completely and becomes A236602.
LINKS
Stanislav Sykora, On Neighbor-Property Cycles, Stan's Library, Volume V, 2014.
FORMULA
For n in A000069, a(n) = 0. - Max Alekseyev, Nov 14 2025
EXAMPLE
The two cycles for n=5 (cycle length 10) are: C_1={1,3,7,5,4,6,2,10,8,9}, C_2={1,5,4,6,7,3,2,10,8,9}.
MATHEMATICA
A242530[n_] := Count[Map[lpf, Map[j1f, Permutations[Range[2, 2 n]]]], 0]/2;
j1f[x_] := Join[{1}, x, {1}];
btf[x_] := Module[{i},
Table[DigitCount[BitXor[x[[i]], x[[i + 1]]], 2, 1], {i,
Length[x] - 1}]];
lpf[x_] := Length[Select[btf[x], # != 1 &]];
Table[A242530[n], {n, 1, 5}]
(* OR, a less simple, but more efficient implementation. *)
A242530[n_, perm_, remain_] := Module[{opt, lr, i, new},
If[remain == {},
If[DigitCount[BitXor[First[perm], Last[perm]], 2, 1] == 1, ct++];
Return[ct],
opt = remain; lr = Length[remain];
For[i = 1, i <= lr, i++,
new = First[opt]; opt = Rest[opt];
If[DigitCount[BitXor[Last[perm], new], 2, 1] != 1, Continue[]];
A242530[n, Join[perm, {new}],
Complement[Range[2, 2 n], perm, {new}]];
];
Return[ct];
];
];
Table[ct = 0; A242530[n, {1}, Range[2, 2 n]]/2, {n, 1, 10}] (* Robert Price, Oct 25 2018 *)
PROG
(C++) // See Sykora link.
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Stanislav Sykora, May 30 2014
EXTENSIONS
a(16)-a(20) from Fausto A. C. Cariboni, May 10 2017, May 15 2017
a(21)-a(22) from Max Alekseyev, Nov 14 2025
a(23) from Martin Fuller, Mar 10 2026
STATUS
approved
