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.

\[ F_n = F_{n-1} + F_{n-2} \]

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.