I Continued fractions and nested radicals

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TL;DR Summary
Isomorphism between continued fractions and nested radicals.
There appears to be a simple isomorphism between continued fractions and nested radicals.

Does anybody know more about this?
 

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y=\sqrt{x+\sqrt{x+\sqrt{x+...}}}
From your observation
y^2-y-x=0
y=\frac{1}{2}(1 \pm \sqrt{1+4x})
as x>0,y>0
y=\frac{1}{2}(1+\sqrt{1+4x})
But from the first formula, y(x=0) should be zero. How can we get value of y(x=1) from it which does not show us initial value ? I am afraid this formula is not defined well enough.
 
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