Patterns in the repetend in $1/121$

Question

$$ \frac{1}{121} = 0.00\ \overbrace{8264}\ \overbrace{4628}\ 09\ \overbrace{91735}\ \overbrace{53719} \ldots $$ The entire $22$-digit repetend appears here. It begins with the first digit after the decimal point. The sequence $8264$ gets reversed and appears as $4628$, and then the same happens with $91735$. But the $09$ between those two doesn't fit any such pattern that I've noticed.

Can anything intelligent be said about this? Is it an instance of some phenomenon that has other instances? Where is it mentioned in the literature?

Answer 1

Try looking at it like this: $$ \frac{1}{121} = 0.0\ \overbrace{08264}^a\ \overbrace{46280}^\bar a\ 9\ \overbrace{91735}^b\ \overbrace{53719}^\bar b\ 0\ 08264 \ldots $$ It becomes much more obvious what's going on if you look at it with this patterning. Notice that $a+b=99999$ and $\bar a+\bar b=99999$, while $a$ and $\bar a$, and $b$ and $\bar b$, are digit-reversals of each other. The remaining two digits sum to 9, naturally.

Answer 2

Interesting, http://www.wolframalpha.com/input/?i=1%2F121. From the "more digits" tab, it looks as though the pattern begins with 00, then a 4 digit sequence and the reverse concatenation, and alternates to 09, followed by a 5 digit sequence and the reverse concatenation.

So the pattern suggests a sequence of the form 00(++++)(----)09(+++++)(-----)00... where the plus signs are the digit sequence and the minus signs are the reverse. The sequence (without it's reverse) after the 00 have an even numbered sequence of even digits while the sequence after 09 have an odd numbered sequence of odd digits.

very lovely, no clue as to why though.