Let
and
be opposite vertices of an octagon. A frog starts at vertex
From any vertex except
it jumps to one of the two adjacent vertices. When it reaches
it stops. Let
be the number of distinct paths of exactly
jumps ending at
. Prove that:
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Let $A$ and $E$ be opposite vertices of an octagon. A frog starts at vertex $A.$ From any vertex except $E$ it jumps to one of the two adjacent vertices. When it reaches $E$ it stops. Let $a_n$ be the number of distinct paths of exactly $n$ jumps ending at $E$. Prove that: $$a_{2n-1}=0, \quad a_{2n}={(2+\sqrt2)^{n-1} - (2-\sqrt2)^{n-1} \over\sqrt2}.$$
Školjka