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If this ends up being a ridiculous question I will delete it. Forgive me if this is ridiculous but this number has me stumped.

$$1.52360679774998$$

The continued fraction calculator gives $1, 1, 1, 10, 11, 11, 11, 11, 11, 11 ...$ which makes me think this number should have a nice expression as a root or log of something or be related to some special number like $\phi$. But I've been unable to tease out any such expression. I appreciate it if someone has more insight into this.

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1  
Wolfram Alpha gives possible closed form solutions, if that's what you're looking for. (e.g., $$ \frac{1}{10}\left(13+\sqrt{5}\right) $$) –  Envious Page Sep 8 '12 at 4:23
    
What are the rest of the digits? Where did this number come from? –  Qiaochu Yuan Sep 8 '12 at 4:24
    
@EnviousPage: Snap. I knew it had something to do with $\phi$. If you had posted this as an answer the check would have gone to you. –  ben Sep 8 '12 at 4:34

3 Answers 3

up vote 5 down vote accepted

If we assume that the continued fraction expansion that you quoted continues in that pattern forever, we can do the calculation by hand. For let $x$ be the value of the continued fraction $\langle 0;11,11,11,\dots\rangle$. Then $x=\dfrac{1}{11+x}$. This gives a quadratic equation with positive root $\dfrac{5\sqrt{5}-9}{2}$.

Now we can claw our way to the top. For example, $\langle 0;10,11,11,11,\dots\rangle=\dfrac{1}{10+x}=\dfrac{2}{5\sqrt{5}+9}$. Continue, resisting the urge to rationalize the denominator. At the end we get $\dfrac{5\sqrt{5}+31}{10\sqrt{5}+20}$. Finally, rationalize the denominator. We get $\dfrac{13+\sqrt{5}}{10}$.

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Answer: $$\frac{13+\sqrt{5}}{10} \approx 1.52360679774997896964091736687...$$

(I got the result using the Inverse Symbolic Calculator: http://isc.carma.newcastle.edu.au )

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I get a 404 with your link, but deleting advancedCalc gets me there –  Ross Millikan Sep 8 '12 at 4:30
    
Try this link. –  Brian M. Scott Sep 8 '12 at 4:31

As I had said in the comments, Wolfram Alpha gives multiple closed form solutions for numbers. The first three are:

$$ \frac{1}{10}\left(13+\sqrt{5}\right), \quad \frac{\Phi+7}{5}, \quad \frac{1}{5\Phi}+\frac{6}{5} $$

where $\Phi$ is the golden ratio conjugate.

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