Some Properties of the Fibonacci Sequence on an Infinite Alphabet

Abstract

The infinite Fibonacci sequence $\mathbf{F}$, which is an extension of the classic Fibonacci sequence to the infinite alphabet $\mathbb{N}$, is the fixed point of the morphism $\phi$: $(2i)\mapsto (2i)(2i+1)$ and $(2i+1)\mapsto (2i+2)$ for all $i\in\mathbb{N}$. In this paper, we study the growth order and digit sum of $\mathbf{F}$ and give several decompositions of $\mathbf{F}$ using singular words.