The number of valid factorizations of Fibonacci prefixes

TL;DR

This paper derives recurrence relations and an explicit formula for counting factorizations of Fibonacci prefixes into decreasing Fibonacci words, revealing that the sequence is a shuffle of two linear functions' ceilings.

Contribution

It introduces new recurrence relations and an explicit formula for V(n), the number of factorizations of Fibonacci prefixes, advancing understanding of Fibonacci word factorizations.

Findings

V(n) sequence is the shuffle of two linear functions' ceilings

Derived recurrence relations for V(n)

Provided an explicit formula for V(n)

Abstract

We establish several recurrence relations and an explicit formula for V(n), the number of factorizations of the length-n prefix of the Fibonacci word into a (not necessarily strictly) decreasing sequence of standard Fibonacci words. In particular, we show that the sequence V(n) is the shuffle of the ceilings of two linear functions of n.