If we have some approximation $x$ for $\pi$, it is possible to improve that approximation by calculating $\sin(x) + x$ if $x$ is sufficiently close to $\pi$. The reason why this works is that for $x \approx \pi$, $\sin(x) \approx \pi - x$ (note that $\sin'(\pi) = -1$), so $x + \sin(x) \approx x + \pi - x = \pi$.

I am interested in the number of good digits when approximating $\pi$ by iteratively applying this technique iteratively starting with the number $3$. In other words, I am interested in the following sequences:

$$ a_0=3; a_{n+1}=\sin(a_n)+a_n\\ b_n=\text{The number of digits of accuracy of }a_n $$

The first few elements of $b$ are $\{0, 3, 10, 32, 99, 300, 902, 2702\}$. I did not find this sequence in OEIS. Interestingly, the number of correct digits seems to *almost* triple with every step.

**Why does this method of approximating $\pi$ triple the number of accurate digits?** If this approximation or sequence has been studied before, any pointers are welcome as well.