Sequences generated from the Dragon Curve

Given the Dragon curve A014577, with offset 0. Let the new sequence S(n) have an offset of (-1), say 1. Then the next term = twice current if A014577 = 1. If A014577 = 0, next term of S(n) = (1/2) current term:
....1...1...0...1...1...1...0...; = A014577
1..2...4...2...4...8...4...2... = S(n).

Alternatively, take the first 2^n terms of S(n). Reverse, double, and append. Thus
given (1, 2, 4, 2), reverse getting (2, 4, 2, 1); double = (4, 8, 4, 2) and append, getting:
(1, 2, 4, 2, 4, 8, 4, 2...) as above.

The A014577 code may define any operation as to (0,1); such as triple, add, or take powers.
The sequence based on the doubling (or halving) operation; is now A166242 in OEIS