Is the continued fraction $\left[1;2,2,3,3,3,4,4,4,4\dots\right]$, where every positive integer $n$ is repeated $n$ times in order starting at $1$, a known value? What properties does it have? I've been unable to find any reference to this number anywhere (with an admittedly limited search).
The only observation I've been able to make is that its slightly less than $\sqrt2$, which makes sense as the continued fraction expansion of $\sqrt2$ is $[1;\bar2]$, and the above series diverges from that one quite slowly (and the elements of the series have less influnce over the final value the later into the series they are).