Question 156

ABCDEFGH is a regular octagon. A and E are opposite vertices of the octagon. A frog starts jumping from vertex to vertex, beginning from A. From any vertex of the octagon except E, it may jump to either of the two adjacent vertices. When it reaches E, the frog stops and says there. Let $$a_n$$ be the number of distinct paths of exactly n jumps ending in E. Then, what is the value of $$a_{2n-1}$$?


The number of vertices between A and E is 3. So, a minimum of 4 steps is needed for the frog to jump from A to E. Also, the frog can go to E from A along any path in only an even number of steps. (2n - 1) is odd. So, the frog can never reach E from A in (2n-1) number of steps. So, the answer is 0

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