ℚ-bonacci Words and Numbers
Sergey Kirgizov · The Fibonacci Quarterly · 2022
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 0a1b satisfy a = 0 or aq > b. When q is an integer we rediscover classical multistep 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.