Positions of Instances of b in Fibonacci String

Theorem

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

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

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

Then the $k$th letter of $S_n$ is $\text b$ if and only if:

$\left\lfloor{\left({k + 1}\right) \phi^{-1} }\right\rfloor - \left\lfloor{k \phi^{-1} }\right\rfloor = 1$

where:

$\left\lfloor{\, \cdot \,}\right\rfloor$ denotes the floor function

$\phi$ denotes the golden mean.

Proof

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