Playing around with numbers, I conjectured three incredibly interesting things:

$$9+\cfrac{1}{18+0\times 12\cfrac{1}{18+1\times 12+\cfrac{1}{18+2\times 12+\cfrac{1}{18+3\times 12+\ddots}}}}=\frac{4e^{1/3}-2}{e^{1/3}-1}$$

$$6+\cfrac{1}{9+0\times 6+\cfrac{1}{9+1\times 6+\cfrac{1}{9+2\times 6+\cfrac{1}{9+3\times 6+\ddots}}}}=\frac{4e^{2/3}-2}{e^{2/3}-1}$$

$$5+\cfrac{1}{6+0\times 4+\cfrac{1}{6+1\times 4+\cfrac{1}{6+2\times 4+\cfrac{1}{6+3\times 4+\ddots}}}}=\frac{4e-2}{e-1}$$

So, what in God's name is going on behind the scenes? Why does this seem to be true, why does it involve $e$, so many questions! All I did was play around on a calculator with some continued fractions, went to Wolfram Alpha and asked for the result to be written in terms of $e$ and then I noticed some patterns. But what is *really* going on? Well, apart from a bit of luck, I have no idea.

Any ideas? Thanks.

**Edit:**

This question may be of help, as it reveals general continued fractions regarding the hyperbolic tangent, which to those who don't know, is a function with respect to some value $\alpha$ defined as $\tanh(\alpha):=\frac{e^{2\alpha} -1}{e^{2\alpha}+1}.$