
COMMENTS

By a canonical Gray cycle (CGC) of length 2n is intended a monocyclic permutation of the integers {0,1,2,...,2n1} 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,...,(2n1), 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: A066037(n)=A236602(2^(n1)).


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 *)
