A236602 Number of canonical Gray cycles of length 2n.

1, 1, 1, 6, 4, 22, 96, 1344, 672, 3448, 12114, 158424, 406312, 4579440, 37826256, 906545760, 362618304, 1784113248, 5576251408, 68998853808, 154939065862

Offset: 1

Author

Stanislav Sykora, Feb 01 2014

Keywords

• nonn
• hard
• more

Comments

By a canonical Gray cycle (CGC) of length 2n is intended a monocyclic permutation of the integers {0,1,2,...,2n-1} such that (i) it starts with "0", (ii) the binary expansions of any two adjacent terms of the cycle differ by exactly one bit, and (iii) the last term is larger than the second. Note: there are no CGC's of odd length.

For n>1, a(n) is also the number of all distinct Hamiltonian circuits in a simple graph with 2n vertices, labeled 0,1,2,...,(2n-1), in which two vertices are connected by an edge only if the binary expansions of their labels differ by exactly one bit.

The sequence is a superset of A066037.

Examples

			a(5) = 4 since there are only these 4 CGC's of length 10:
{0 2 3 7 6 4 5 1 9 8}
{0 2 6 4 5 7 3 1 9 8}
{0 4 5 7 6 2 3 1 9 8}
{0 4 6 2 3 7 5 1 9 8}
		

Links

• Stanislav Sykora, On Canonical Gray Cycles, Stan's Library, Vol.V, January 2014, DOI: 10.3247/SL5Math14.001.

Crossrefs

Cf. A066037 (subset), A236603, A350784.

Programs

Mathematica

A236602[n_] := Count[Map[lpf, Map[j0f, Permutations[Range[2 n - 1]]]], 0]/2;
j0f[x_] := Join[{0}, x, {0}];
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 &]];
Join[{1}, Table[A236602[n], {n, 2, 5}]]
(* OR, a less simple, but more efficient implementation. *)
A236602[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[]];
      A236602[n, Join[perm, {new}],
       Complement[Range[2 n - 1], perm, {new}]];
      ];
     Return[ct];
     ];
   ];
Join[{1}, Table[ct = 0; A236602[n, {0}, Range[2 n - 1]]/2, {n, 2, 8}] ](* Robert Price, Oct 25 2018 *)

Formula

a(2^(n-1)) = A066037(n).

a(n) = A350784(n)/2 for n >= 2. - Martin Ehrenstein, Feb 16 2022

Extensions

a(17)-a(18) from Fausto A. C. Cariboni, May 13 2017

a(19)-a(20) from Martin Ehrenstein, Feb 16 2022

a(21) from Martin Ehrenstein, Feb 21 2022