Let's look at φ^1, now! Anything to the first power is just that number, so φ^1=φ! Think of that as φ^1 = φ + 0

(because adding 0 does nothing)

Remember our Fibonacci numbers, now. Starts out with 0, 1, 1, 2, etc. Remember that F_0=0 and F_1 = 1?

φ^1 = φ + 0

can be rewritten as

φ^1 = F_1 * φ + F_0

Which means that φ^1 follows that formatting that we used earlier! Thus, every φ^n follows the format

φ^n = F_n * φ^n + F_(n-1)

that's nifty

Anyways I'll stop being a Big Nerd for like 2 minutes