Golden ratio, $n$-bonacci numbers, and radicals of the form $\sqrt[n]{\frac{1}{n-1}+\sqrt[n]{\frac{1}{n-1}+\sqrt[n]{\frac{1}{n-1}+\cdots}}}$

This is equivalent to $x^n=\dfrac1{n-1}+x$ , which means solving a polynomial of the n-th degree in x. The generalized Fibonacci numbers fulfill the general equation $x^{−n}+x=2$, so there is no reason to expect any universally valid connection between the two. See also strong law of small numbers.