Non-standard quaternary representations and the Fibonacci numbers

2025/05/07 by Anders, Katie, Dawsey, Madeline L., Gupta, Rajat +2
#11A63 #11B37 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2505.04589

Abstract

Let f₄(n) be the number of hyperquaternary representations of n and b₄(n) be the number of balanced quaternary representations of n. We show that there is no integer k such that f₄(n+k)=b₄(n) for all n≥ -k, in contrast to the binary case. Nevertheless, there do exist integers k such that f₄(n+k)=b₄(n) for arbitrarily large intervals of n. We generalize these results to any even base d. We also study the rate of growth of b₄(n) and show that maximal values of this function correspond to certain Fibonacci numbers.

Citations

• Non-standard binary representations and the Stern sequence
• The maximal order of hyper-(b-ary)-expansions
• Upper Bounds for Stern's Diatomic Sequence and Related Sequences
• Congruence properties of binary partition functions

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