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arXiv · 2201.00782

$\mathbb{Q}$-bonacci words and numbers

Sergey Kirgizov
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Abstract

We present a quite curious generalization of multi-step Fibonacci numbers. For any positive rational $q$, we enumerate binary words of length $n$ whose maximal factors of the form $0^a1^b$ satisfy $a = 0$ or $aq > b$. When $q$ is an integer we rediscover classical multi-step Fibonacci numbers: Fibonacci, Tribonacci, Tetranacci, etc. When $q$ is not an integer, obtained recurrence relations are connected to certain restricted integer compositions. We also discuss Gray codes for these words, and a possibly novel generalization of the golden ratio.

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Sergey Kirgizov. 2022-07-14. $\mathbb{Q}$-bonacci words and numbers. https://arxiv.org/abs/2201.00782

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