Most likely square for a one-or-two jumping frog

A frog sits at $0$ on the number line and jumps forward repeatedly, each jump being $1$ or $2$ units with probability $\tfrac12$ each. For $k>0$, let $p_k$ be the probability the frog ever lands exactly on $k$. What is the maximum of $p_k$ over all $k>0$?

$3/4$. Position $k$ is reached from $k-1$ or $k-2$, so $p_k=\tfrac12p_{k-1}+\tfrac12p_{k-2}$ with $p_0=1$, $p_1=\tfrac12$. Solving gives $p_k=\tfrac23+\tfrac13\left(-\tfrac12\right)^k$, which is largest at $k=2$: $\tfrac34$.