Right! Let's spell it out: a path \$$\gamma: x \to x\$$ of length \$$n \ge 2 \$$ must either be of the form \$$f \circ \delta\$$ for some path \$$\delta : x \to x\$$ of length \$$n-1\$$, or of the form \$$h \circ g \circ \delta \$$ for some path \$$\delta : x \to x\$$ of length \$$n-2\$$. So, if \$$F_n\$$ is the number of paths of length \$$n\$$, we must have

$F_n = F_{n-1} + F_{n-2}$

for \$$n \ge 2\$$, but also it's easy to see \$$F_0 = F_1 = 1\$$. Fibonacci!

Another way to say it is that the Fibonacci number \$$F_n\$$ counts the number of ways to write \$$n\$$ as an _ordered_ sum of 1's and 2's. For example

$4 = 1 + 1 + 1 + 1 = 2 + 1 + 1 = 1 + 2 + 1 = 1 + 1 + 2 = 2 + 2$

so the fourth Fibonacci number is 5.