Positions of Instances of a in Fibonacci String

Theorem

Let $S_n$ denote the $n$th Fibonacci string.

Let $F_n$ denote the $n$th Fibonacci number.

Let $m \in \Z$ such that $m \le F_n$.

Let $m - 1$ be expressed in Zeckendorf representation as $Z_{m - 1}$.

Then the $m$th letter of $S_n$ is $\text a$ if and only if:

$k_r = 2$

where $k_r$ denotes the final digit of $Z_{m - 1}$.

Proof

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