0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1

0 -> 1   01 -> 10   0110 -> 1001   01101001 ->  10010110
Recursively append the inverse.

This sequence is known as the Morse-Thue sequence.  It can be defined
non-recursively as the nth term is the mod 2 count of 1s in n written
in binary:
        0->0 1->1 10->1 11->0 100->1 101->0 110->0 111->1 etc. 

Reference:
    Dekking, et. al., "Folds! I,II,III"
    The Mathematical Intelligencer, v4,#3,#4,#4.