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@syockit, indeed there's an error, thank you for pointing that out. Instead of $x_n$ it should be the values $y_n$ (see e.g., equation 3.61 of the book I talked about in the answer).
@Whuber you are using Monte Carlo to solve the quadratic equation which is originated in e.g. the branching process proposed by Silverfish's answer. Actually, to be more precise, all the other answers are openly based on self-referencing or self-replicating approaches. Yours is the only one that echoes by using the equation as a starting point.
@Aksakal but the point is the following: how could we ask for the estimation of any of $\Phi$'s irrational peers? Let's say $1.89253...$ is one of those. Would it make sense to simply ask for statistically approaching $1.89253...$? Obviously not. Now, if I ask about $\Phi$, it immediately defines a family of algorithms, and that's the point of using $\Phi$ here. Look at the answers, if they had changed a detail in the algorithm, they would not get $\Phi$, nonetheless, all the answers smell of self-replication and self-referencing. Whether $\Phi$ is special or not is just not the point at all.
@Aksakal $\Phi$ appears in self-replication problems in a similar way in which $e$ appears in exponential growth. It's just one possible value out of an infinite family and is not special with respect to its peers. You claimed that "it is not interesting to think about it in this way". Leaving aside that what is interesting or not is very relative, the point is that estimating $\Phi$ statistically brings up very different kind of approaches from those used to estimate $\pi$ or $e$ (see my question), and that's what makes this interesting, not whether we aim for $\Phi$ or for $1.89253$.
