First of all, I apologize for my amateurness and inexperience. Although I always enjoyed math, only two years ago I started experimenting with continued fractions and gained a deep reverence for them.
I first "discovered" the trivial but beautiful identities linking the golden ratio to the n-th Lucas number through raw experimentation:
$φ^n = L_n-\cfrac{1}{L_n-\cfrac{1}{L_n-\cfrac{1}{L_n-\cdots}}} $ $, n∈2N$
$φ^n = L_n+\cfrac{1}{L_n+\cfrac{1}{L_n+\cfrac{1}{L_n+\cdots}}}$ $, n∈2N+1$
$\lim_{n\to\infty} ϕ^n=L_n$
Which follows from $\phi^n + \phi^{-n} = L_n$ $, n∈2N$
$\phi^n - \phi^{-n} = L_n$ $, n∈2N+1$
I gained great enjoyment from messing about with numbers but unfortunately starting university had distracted me until recently, when right before I slept I somehow jotted down the following trivial but "beautiful" results:
$e = 3-\cfrac{1}{3+\cfrac{1}{2-\cfrac{1}{5+\cfrac{1}{2-\cfrac{1}{7+\cfrac{1}{2-\cfrac{1}{9+\cdots}}}}}}} $
$e = 1+\cfrac{1}{1-\cfrac{1}{2+\cfrac{1}{3-\cfrac{1}{2+\cfrac{1}{5-\cfrac{1}{2+\cfrac{1}{7-\cdots}}}}}}} $
While I realize all of these results have been probably published hundreds of years ago, the process of independently finding them out is extremely enjoyable to me. I have become obsessed with mathematics! I love it and often spend so much time reading and experimenting that I fall asleep exhausted. I would like to progress further with mathematics. Would any of the more experienced members have any recommended introductory texts on the topic of continued fractions and mathematics in general? Thank you for your patience and time reading this.