The “Fibonacci sequence” is defined as a sequence of numbers such that you have the recursion: , and the restrictions: and . Explicitly, the Fibonacci sequence is: 1, 1, 2, 3, 5, 8, 13, 21, … That is, the recursion says that every term is the sum of the previous two. This derivation is for the ordinary sequence, but it can be altered to suit any generalized Fibonacci sequence. So far, using what is known about the Fibonacci sequence, we’ve found a nice closed equation for the generating function (g), which “encodes” the sequence. Hopefully, we can use this fancy new equation to figure out what each f n must be.

When you want to outsmart the world, you turn to the facts. And the facts are in the science.