Positions of Instances of a in Fibonacci String
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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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Sources
- 1976: Kenneth B. Stolarsky: Beatty sequences, continued fractions, and certain shift operators (Canadian Math. Bull. Vol. 19: pp. 473 – 482)
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.8$: Fibonacci Numbers: Exercise $36$