Such sequences exist and are called De Bruijn sequences. One way of creating them is to insert a 0 into the longest run of zeroes in one period of the output of a maximal length LFSR. A De Bruijn sequence of period 2^n can be generated by a n-stage feedback shift register with _nonlinear_ feedback.
The longest run of zeroes from a n-stage maximal-length LFSR is n-1 consecutive zeroes. This is because if the LFSR ever gets into the all-zeroes state it would continue in that state forever and produce a sequence of zeroes only). The nonlinear feedback actually is just the feedback that occurs in the LFSR **except** that a 0 is fed back instead of a 1 when the register contents are
000...01 (100...00 for you Republicans) so that the next contents are 000...00 (regardless of which party you belong to!) instead of 100...00 (or 000...01) as would be the case with a LFSR. Nonlinear feedback strikes again when the shift register contents are 000...00, feeding back a 1 instead of a 0 so that the next contents are 100...00 (or 000...01) and the linear feedback resumes. In short, if we draw a state transition diagram for the De Bruijn register, it looks very much like the state transition diagram for the LFSR except for the notable difference that the 000...00 state is inserted between 000...01 and 100...00 (and it needs nonlinear feedback to accomplish this).Dilip Sarwate