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

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.