Abstract:
In this paper we explore combinatorial properties of Fibonacci words and their generalizations within the framework of combinatorics on words. The paper measures the diversity of subwords in Fibonacci words, showing non-decreasing growth for infinite sequences. We extend factor analysis to arithmetic progressions of symbols, highlighting generalized pattern distributions. Recent results link Sturmian sequences (including Fibonacci words) to unbounded binomial complexity and gap inequivalence, with implications for formal language theory and automata.
In this work, the infinite word $\mathfrak{F}=\mathfrak{F}_{b}:=\left({ }_{b} f_{n}\right)_{n \geqslant 0}$ is defined by concatenating non-negative base- $b \geqslant 2$ representation of the recursive $n$!.
Keywords:density, Fibonacci word, ergodic theory, sequence.
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