Non-standard quaternary representations and the Fibonacci numbers

TL;DR

This paper explores the relationships between hyperquaternary and balanced quaternary representations of integers, revealing unique properties and growth behaviors linked to Fibonacci numbers, and extends findings to even bases.

Contribution

It demonstrates the non-existence of a universal shift aligning hyperquaternary and balanced quaternary representations, and generalizes results to all even bases, highlighting Fibonacci connections.

Findings

No universal integer shift exists for all n in hyperquaternary and balanced quaternary representations.

Existence of large intervals where shifted representations coincide.

Maximal values of the balanced quaternary count relate to Fibonacci numbers.

Abstract

Let $f_4(n)$ be the number of hyperquaternary representations of $n$ and $b_4(n)$ be the number of balanced quaternary representations of $n$. We show that there is no integer $k$ such that $f_4(n+k)=b_4(n)$ for all $n\ge -k$, in contrast to the binary case. Nevertheless, there do exist integers $k$ such that $f_4(n+k)=b_4(n)$ for arbitrarily large intervals of $n$. We generalize these results to any even base $d$. We also study the rate of growth of $b_4(n)$ and show that maximal values of this function correspond to certain Fibonacci numbers.