And I guess a direct formula for circle $$O_n$$ could be
$$
d(n) = \frac{1}{n(n+1)}.
$$

south_korea_ln

Yes, exactly. 

Related to [Apollonian gaskets](https://en.wikipedia.org/wiki/Apollonian_gasket)
> a fractal generated by starting with a triple of circles, each tangent to the other two, and successively filling in more circles, each tangent to another three.

science

[Daily puzzle] Diameter of circle

Oh, neat, I didn't know about Descartes' theorem!

So you're calculating
$$
k_4 = k_1 + k_2 + k_3 + 2*\sqrt{k_1*k_2 + k_2*k_3 + k_3*k_1}
$$

where curvature $$k_n$$ is $$\frac{1}{r_n}$$ (the inverse of the radius).

If I pick $$O$$, $$O'$$ and $$O_1$$ to calculate $$O_2$$

$$
1 + 1 + 4 + 2*\sqrt{1*1 + 1*4 + 4*1} = 12
$$

Now we can simply repeat with $$O$$, $$O'$$ and $$O_2$$ to calculate $$O_3$$

$$
1 + 1 + 12 + 2*\sqrt{1*1 + 1*12 + 12*1} = 24
$$

So radius $$\frac{1}{24}$$ or diameter $$\frac{1}{12}$$
