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We know

$$\frac{1+\sqrt{5}}{2} = 1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cdots}}}$$

and

$$e = 2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\frac{1}{4+\cdots}}}}.$$

There are similar formulas for any quadratic irrationals and some other numbers related to $e$ (such as some rational powers and $\frac{2}{e-1}$, though those can be quite rich, and I would be interested in any remarkable examples). A bit less known is

$$\frac{I_1(2)}{I_0(2)} = \cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{3+\cdots}}},$$

where $I_{\alpha}$ is the modified Bessel function. In contrast, many numbers such as $\pi$ and algebraic irrationals of degree $>2$ are expected to have random-looking continued fractions.

Are there any other examples of numbers whose representation as standard continued fraction is "known" (or depending on how you look at it, any other continued fractions we can compute)?

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$$\tanh(1/k) = \dfrac{1}{k + \dfrac{1}{3k + \dfrac{1}{5k+\dfrac{1}{7k + \ldots}}}}$$ and $$ \tan(1/k) = \dfrac{1}{k-1+ \dfrac{1}{1+\dfrac{1}{3k-2 + \dfrac{1}{1+\dfrac{1}{5k-2 + \dfrac{1}{1+\ldots}}}}}}$$

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    $\begingroup$ Thanks for these formulas, though I would classify them as "numbers related to e" (e.g. $tan(1/k) = (e^{2i/k}-1)/(e^{2i/k}+1)$) $\endgroup$
    – Rodrigo
    Dec 15, 2019 at 3:21

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