On three conjectures of Kimberling concerning the array $\lfloor kφ^n\rfloor$

Abstract

Let $φ$ be the golden ratio and let $R_n=\{\lfloor kφ^n\rfloor : k\ge 1\}$ be the $n$-th row of the array $T(n,k)=\lfloor kφ^n\rfloor$ (OEIS A128440). In 2022 Kimberling conjectured that the rows $R_{2n-1}$ and $R_{2n}$ are disjoint, and that after the two rows are merged and each entry is replaced by its rank, they become the lower and upper Wythoff sequences. He also conjectured (OEIS A358359) that if $a(N)$ is the number of rows containing $N$, then every positive integer occurs infinitely often among the values of $a$. We show that the first two conjectures follow quickly from the Skolem-Bang theorem, which also yields the exact rule for when two rows are disjoint: $R_i\cap R_j=\emptyset$ ($i<j$) if and only if $j-i$ is odd and divides $i$. We then prove the third conjecture. The main tools are an explicit determination of the rows containing an odd-indexed Lucas number, which extends a result of Noppakaew, Kanwarunyu and Wanitchatchawan, and a "Lucas shift" lemma: if $N+1$ is not of the form $L_{2e}$ with $e\ge1$, then adding a sufficiently large even-indexed Lucas number to $N$ does not change the set of rows containing it. We also show that each value of $a$ is taken on a set of positive natural density, and we report computations up to $10^8$ suggesting that the least $N$ lying in exactly $v\ge 2$ rows is the Lucas number $L_{4v-5}$.