Hard square root and binary representation question

If $\frac{1}{\sqrt{2}}$ is representend in binary the value is $0.10110101000001...$ How can we represent the number above after inserting a zero between each digit ($0.1000101000100010000000000010$) with a formula (function)? (this formula should contain decimals constans)

EDIT: If such formula (function) doesn't exist, how to prove it?

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Why would you think such a "formula" exists? – Eckhard Dec 27 '12 at 14:59
I will edit that. But if such formula doesn't exist. Is there a way to prove that? – user1897330 Dec 27 '12 at 15:00
What is your definition of a formula – Amr Dec 27 '12 at 15:04
I mean a function. – user1897330 Dec 27 '12 at 15:05

For any $a \in [0,1)$, we can define a unique analytic function $f_{a}(z)$ (over at least the open unit disk) by $$f_{a}(z)=\sum_{i=0}^{\infty}a_{i}z^{i},$$ where the binary representation of $a$ is $0.a_0a_1a_2...$ Then $(1/2)f_{a}(1/2)=a$, and $(1/4)f_{a}(1/4)$ is equal to the "spaced out" version of $a$ that you described. Of course, finding any other representation of $f_{1/\sqrt{2}}$ then becomes the trick.