Interesting. I did not know this before but it turns out that fibonacci grows as phi^n where phi is the golden ratio (approximately = 1.61803398874)

1. n^2 is a power law, not an exponential
2. 2^n is exponential, I am not sure why you refer to it as “true exponential”, other than software engineers as a rule don't seem to understand the difference between power law and exponential and are constantly trying to re-invent math. Specially if the company is pre-IPO, when everything is “exponential” or “hockey-stick”.
3. I'm glad you take the log(y), but the way to differentiate between power law and exponential is on a log-log plot: power law is a straight line on a log-log, exponential is straight line on a semi-log plot.
4. In your semi-log plot, the Fibonacci yellow line is simply hidden under the blue.
5. As Don points out, Fibonacci grows asymptotically as exponential. Proof: start with the definition of Fibonacci. Divide by the middle term, assume that the ratio of sequential terms converges for large term number, this yields the definition of the Golden ratio, namely that t(n)/t(n-1) ~ phi. Hence t(n) ~ phi**n.

When a 50 word blog post starts with “a random neuron fired” it's probably safe to say that a Field Medal proof won't be coming next.

So thanks I guess?

Lots of useless shade in the comments. These graphs are exactly what I needed to see. Thank you!