ℚ-bonacci words and numbers

01/03/2022

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by   Sergey Kirgizov, et al.

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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.