numerical values of points in cantor set

Question

Let $C$ be the standard middle thirds Cantor set in the interval $[0,1]$. The "endpoints" of $C$ have very simple numerical values that can be listed off: $$0,1,1/3,2/3,1/9,2/9,6/9,7/9,1/27,2/27,7/27,8/27,... $$

What I am looking for is some numerical values of the "nonendpoints" of $C$. Specifically, is there a dense subset of the nonendpoints which has some nice pattern to its values? Can you list off some of these for me?

Answer 1

For example,

$$\frac 14 = \frac{0}{3} + \frac{2}{3^2} + \frac{0}{3^3} + \frac{2}{3^4} + \frac{0}{3^5} + \frac{2}{3^6} + \cdots$$

is in the Cantor Set.

Edit: Actually, elements of the Cantor Set are of the form

$$\sum_{n=1}^\infty \frac{a_n}{3^n}$$

where $a_n$ is any sequence consisting of only $0$ and $2$'s.

So, when $a_n$ is not almost constant (an almost constant sequence is a sequence which is constant from some point on) then the corresponding number is a non-endpoint.

Edit2:

$$\frac{2}{25} = \frac{0}{3} + \frac{0}{3^2} + \frac{2}{3^3} + \frac{0}{3^4} + \frac{0}{3^5} + \frac{2}{3^6} + \cdots$$

$$\frac{11}{12} = \frac{2}{3} + \frac{2}{3^2} + \frac{0}{3^3} + \frac{2}{3^4} + \frac{0}{3^5} + \frac{2}{3^6} + \cdots$$

For further examples, you can take:

$a_n = 0,0,0,2,0,2,0,2,0,2,0,2,0, \dots$

$a_n = 2,0,2,0,2,2,0,2,0,2,0,2,0, \dots$

$a_n = 0,2,2,0,0,2,2,0,0,2,2,0,0, \dots$

$a_n = 0,0,0,0,2,2,2,2,0,0,0,0,2,2,2,2, \dots$

$a_n = 0,0,0,2,2,0,0,0,2,2,0,0,0,2,2, \dots$

etc.