The leap-frog's favorite landing spot

HardProbability Stochastic Processes~8m

A frog starts at position $0$ on the integer line. At each step it jumps forward $1$ or $2$ units, each with probability $\tfrac12$ . Let $p_k$ be the probability the frog ever lands exactly on position $k$ (for $k > 0$ ). What is the largest value of $p_k$ over all $k > 0$ ?

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Reach $k$ from $k-1$ (then $+1$ ) or $k-2$ (then $+2$ ): $p_k=\tfrac12 p_{k-1}+\tfrac12 p_{k-2}$ , $p_0=1$ , $p_1=\tfrac12$ .
The values oscillate toward $\tfrac23$ ; the peak is $p_2=\tfrac34$ .

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0.75 (± 0.001)

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The leap-frog's favorite landing spot

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