I was working on a self made Infinite Resistor Network Problem in Physics.
I have removed the Physics part and only left the Mathematics here.
Let us start with the $n$th Fibonacci Number $F_n$, our notation would be such that $F_1=1$.
Now we have to apply a set of algorithms on $F_n$.
Let us say $k=F_{n}$ such that $n$ is even.
Then perform the following operations on $k$.
$$
\boxed{
\begin{align}
k&\to k+F_{n-1}, n\to n-1\\
k&\to \frac{kF_{n-1}}{k+F_{n-1}},n\to n-1
\end{align}}$$
We have to keep repeating this process indefinitely until we hit $F_1$.
I think my explanation might not be sufficient so I will show an example for $F_6$.
$$\frac{\left(\frac{\left(5+8\right)\left(3\right)}{\left(5+8\right)+\left(3\right)}+\left(2\right)\right)\left(1\right)}{\left(\frac{\left(5+8\right)\left(3\right)}{\left(5+8\right)+\left(3\right)}+\left(2\right)\right)+\left(1\right)}+\left(1\right)$$
As we are only interested in the even cases, I will put up some partial sums for it. $$2,\frac{11}{6},\frac{158}{87},\frac{2966}{1635}$$
I believe the algorithm converges for $$\lim_{n\to\infty}F_n$$ Numerical evaluations gives us this, $$1.8137926663157207$$
My question is to find a closed form for this value.