$N$-factor complexity of the infinite Fibonacci sequence and digital sequences

TL;DR

This paper introduces the $N$-factor complexity to analyze sequences over infinite alphabets, precisely evaluates it for the infinite Fibonacci sequence, and explores digit sequences based on block occurrences in base-$k$ representations.

Contribution

It defines the $N$-factor complexity for infinite sequences and provides exact calculations for the infinite Fibonacci sequence and certain digit sequences.

Findings

Exact $N$-factor complexity for the infinite Fibonacci sequence.

Analysis of digit sequences based on block occurrences in base-$k$ representations.

Characterization of sequence complexity on infinite alphabets.

Abstract

In this paper, we introduce a variation of the factor complexity, called the $N$-factor complexity, which allows us to characterize the complexity of sequences on an infinite alphabet. We evaluate precisely the $N$-factor complexity for the infinite Fibonacci sequence $\mathbf{f}$ given by Zhang, Wen and Wu [Electron. J. Comb., 24 (2017)]. The $N$-factor complexity of a class of digit sequences, whose $n$th term is defined to be the number of occurrences of a given block in the base-$k$ representation of $n$, is also discussed.