Is it well known that for the Pythagorean spiral, $\sum_1^\infty\frac1{a_n+1} = \frac12$?

Here is Sylvester's sequence whose reciprocals are known to sum to $1$. The sequence you have is Sylvester's sequence doubled minus $1$. So your sum of reciprocals summing to $1/2$ is equivalent to Sylvester's reciprocals summing to $1$.

Note how low the index is at OEIS for Sylvester's sequence. This indicates it's been studied a lot.

This is not an answer to Mark Fischler's question, but an approach for a proof that $\sum\limits_{n=1}^\infty\,\dfrac{1}{a_n+1}=\dfrac12$. The idea is to show that $$\frac{1}{a_n+1}=\frac{1}{a_n-1}-\frac{1}{a_{n+1}-1}$$ for every $n=1,2,3,\ldots$. Then, by telescoping the sum, the claim follows.