| What I find fascinating is that there seem to be so many valid ways to generalize the Golden Ratio. As you say, the "metallic means" [1] are quite well-known, and relate to the recurrence relation via: T(n) = m *T(n-1)+ T(n-2), for some constant integer m.
For example, m=1 is the golden ratio, m=2 is the silver ratio,... But one of my other posts [2], generalizes the Golden ratio via the "Harmonious Numbers", as defined by the lagged recurrence, T(n+m) = T(n)+T(n-1), for some constant m.
In this case, m=1 relates to the Golden Ratio, and m=2 relates to the Plastic Number [3]. And then finally, this post explores generalizing it via a completely different perspective, that of "Lagrange Numbers". It seems that we need to 'think outside the box' a litte when generalizing the Golden ratio, as there is not single obvious way to generalise continued fractions. [1] https://en.wikipedia.org/wiki/Metallic_mean [2] http://extremelearning.com.au/unreasonable-effectiveness-of-... [3] https://en.wikipedia.org/wiki/Plastic_number |