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.
翻译:我们提出了一个相当奇怪的多步Fibonacci数字的简单化。对于任何正正合理的美元,我们列举一元长度的二元字数,其最大因子为0 $a1 $ = 0$或 $a = 美元 或 $a = b$。当美元是整数时,我们重新发现传统的多步Fibonacci数字(Fibonacci、Tribonacci、Tetranacci等)。当美元不是整数时,获得的重复关系与某些有限的整数组成有关。我们还讨论了这些单词的灰色代码,以及黄金比的新的概括。