(Notice how the string 2110 becomes 1333, and the even longer 211110 becomes 133333.)
Binary is an especially tricky case. We notice that with the natural numbers in binary, only the _repunits_ are identical in both standard and NyaN. All other naturals become one digit shorter in NyaN compared to standard binary.
In fact, this rule appears to work converting binary to NyaN:
-- Replace the rightmost 0 with 2. -- Drop the leftmost 1. -- Increase all digits in between by 1.
With irrational numbers like pi, we simply ignore the first statement above, since there is no rightmost digit.