The simple Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ..., where each term is the sum of the two preceding terms, inspired me the Fibonacci Machine that you could see below:

Imagine, in this machine, a start position where a=1 and b=1. Next step, b=old_a and a=old_a+old_b, and so on. To infinity, the Golden Mean appears on machine output.

You could imagine now a more complex machine, where the value of each cell is multiplied by a coefficient (k1, k2) before forming the feedback sum. It's the machine that you could see in the next picture:

The results are very interesting. In particular, the case a=1, b=3 and a=1, b=4 looks like a cabalistic calculation: in many cultures 13 and 17 are magic numbers.

However, we can conceive more and more complex machines.

Link:

Leonardo Pisano Fibonacci