Is there any kind of known pattern to $\sqrt 2$ in base 2? Is there any kind of known pattern to $\sqrt 2$ in base 2?
Is there any classification categories for decimal digits of numbers that for example would put $\sqrt 2, \sqrt 3 \cdots \sqrt n$ into separate category than $\sqrt[3] 2, \sqrt[3] 3 \cdots \sqrt[3] n$
Or given a decimal expansion with arbitrary precision, or definition of digit in the nth place (not necessarily decimal expansion) to get a probability of which class of numbers the number is likely to be? e.g. transcedental or algebraic of degree k?     
 A: One expression for the square root of 2 is given by $\sqrt{2} = \sum_{k=0}^{\infty} (2k+1) \binom{2k}{k}  2^{-3k -1} $ which is rather useful for computing the digits exactly. It also means that any pattern in the binary digits of $\sqrt{2}$ should be somewhat related to the the digits of the central binomial coefficients. There's probably a combinatorial interpretation of this.
Incidentally, $(2k+1) \binom{2k}{k}$ is asymptotic to $\frac{2^{2k + 1}}{\sqrt{\pi}}$ so the number of terms affecting the kth digit is linear in k. Also, the greatest power of 2 dividing the central binomial coefficient is known as Gould's sequence, i,e, $\binom{2k}{k} = 2^{2^{h(k)}}$ where $h(k)$ is the hamming weight of k.
A: there is a method to calculate decimal places of root2, probably that could help find a pattern with it. Although this is not a very obvious one, it probably could (very loosely) help to find a pattern.
The method works a bit like dividing, so you have to calculate the next decimal place, then the next.
It does not work on approximations by the way.
http://www.murderousmaths.co.uk/books/sqroot.htm
On this link you will see a few methods, but scroll down past the cyan shortcut box and you will see the one shown by Frank La Fontaine.
I'm sorry that I can't show an actual pattern, but hopefully this helps.
