Jumpy, Patient Frog Named Sally And A Math Problem About Her Life-II

Let $c_k$ be the number of ways leading from $A$ to $C$ , of length $2k$ . Let $b_k$ be the number of ways of length $2k-1$ leading from $A$ to $B$ . Since, in two jumps, there are two ways of returning from $B$ to $B$ and one way of getting to $F$ , $b_{k+1}=3b_k$ .

However, $c_k=b_k$ , meaning that

$c_{k+1}=3c_k$ when $k>0$ . $c_1$ is still $1$ , and by induction, the generalized formula is $c_k=3^{k-1}$ .

We would like to express this in terms of $n$ . As $k$ is equal to half of $n$ (the paths being of length $n=2k$ ), we have that there are $3^{\frac{n}{2}-1}$ ways for an even $n$ . It is impossible to reach $C$ from $A$ in an odd number of jumps.