Interesting pattern in the decimal expansion of $\frac1{243}$

decimal-expansion fractions

There appears to be an interesting pattern in the decimal expansion of $\dfrac1{243}$:

$$\frac1{243}=0.\overline{004115226337448559670781893}$$

I was wondering if anyone could clarify how this comes about?


$\frac{1}{243}=\frac{1}{333}+\frac{10}{8991}$

$\frac{1}{333}=.\overline{003}$

$\frac{1}{8991}=.\overline{000111222333444555666777889}=\frac{111}{998001}=\frac{111}{10^6-2\cdot10^3+1}$

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