Non-standard quaternary representations and the Fibonacci numbers

TL;DR

The paper investigates non-standard d-ary representations for even bases, focusing on hyper- and balanced-quaternary representations when d=4. It proves there is no universal shift $k$ with $f_4(n+k)=b_4(n)$ for all large $n$, in contrast to the binary case, but shows the existence of long intervals where equality holds, and extends these interval results to general even bases $d=2\ell$. The authors establish Stern-like recurrence relations for $f_d(n)$ and $b_d(n)$, and analyze the rate of growth, revealing that maximal values on specific intervals align with Fibonacci numbers. They provide explicit extremal constructions and maxima for $f_4(n)$ and $b_4(n)$ and extend the framework to higher even bases, including a detailed discussion of digit-elimination techniques essential for the general proofs. Overall, the work connects non-standard base representations with Fibonacci-structured extremal behavior and yields a robust generalization to all even bases.

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.

Theorems & Definitions (23)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3: defant2016upper, Proposition 2.1
• Theorem 1.4
• Theorem 2.1
• Theorem 2.2
• proof
• Lemma 2.3
• proof
• Theorem 3.1