Non-standard quaternary representations and the Fibonacci numbers

Katie Anders; Madeline L. Dawsey; Rajat Gupta; Noah Lebowitz-Lockard; Joseph Vandehey

paper

Non-standard quaternary representations and the Fibonacci numbers

arXiv:2505.04589

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 $n \geq - k$ , 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.