Q-bonacci words and numbers

6533b854fe1ef96bd12afa6d

RESEARCH PRODUCT

Q-bonacci words and numbers

subject

[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO][INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM][MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO][INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM]

description

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.

year	journal	country	edition	language
2022-01-04

https://hal.science/hal-03510280

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