The Dragon Curve mapping codes

As shown in Mathworld, the "L" sequence for the Dragaon curve =
(1, 1, 0, 1, 1, 0, 0,....).
This should actually be prefaced with an 0 getting: (0, 1, 1, 0, 1, 1, 0, 0);
where these terms can be derived from the lengths of continued fraction representations of
one-half of the Infinite Farey tree. We use
1
1, 2
1, 2, 3, 2
1, 2, 3, 2, 3, 4, 3, 2
....= such lengths, then use the substitutions: "1" if next term going to the right is higher; otherwise 0.; but letting leftmost term -= 0; thus the 8 bit string =
0, 1, 1, 0, 1, 1, 0, 0;
now we can use these codes to remap any sequence S(n) say (1, 3, 5, 7,...) onto a 2^n x 2^n Gray Code format.; getting
1, 3, 5, 3
3, 1, 3, 5
5, 3, 1, 3
3, 5, 3, 1
...
as shown in A002697, where the sum of these terms = 48.
And we note that A038208 * [1, 3, 5, 7,...] = (1, 8, 48, 256,....) where A038208 =
1
2, 2
4, 8, 4
8, 24, 24, 8
...a symmetrical triangle with special properties relating to biochemistry per comments shown
in A038208.