This refers to a type of impartial game defined as octal games by Berlekamp, Guy and Conway in the first edition of the Winning Ways books.
I noticed that the nim-sequences of $.17$ and $.117$ (first cousin of $.051$) both have a preperiod of $p-1$ for their period $p$ ($p=34$ and $p=48$).
This seems interesting given they have nearly a full cycle before the periodicity starts. Also fairly unusual? I haven't noticed it in other octal games. I assume that both having this property is due to the similarities in the rules of the games?
I was wondering if anyone was able to conjecture why they have preperiod $p-1$ based on the rules of the games? Or is there a reason in general that ultimately periodic sequences might be likely to have preperiod $p-1$?
Perhaps there is nothing interesting to notice?