A binary De Bruijn sequence of index $n$ is a circular sequence $S=a_1 a_2 \dots a_{2^n},$ with $a_i \in \{0,1\},$ and such that each of the $2^n$ binary $n$-uples occurs exactly once in $S.$

Is there an infinite binary sequence $b_1 b_2 b_3 \dots$ such that $b_1 \dots b_{2^n}$ is a binary De Bruijn Sequence of index $n$ for all $n$ ?