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 $n$ ?



Nested De Bruijn Sequences
No. It may as well start with 0 in which case it is forced to be 01 for order 2 and then be 0110 for order 4. In order to include 111 it could only continue to be 01101110 or 01100111 but both of those have 011 twice and miss 000.


