5 papers · 1 filter
The Tribonacci constant and finite automata
Jeffrey Shallit
We show that there is no automaton accepting the Tribonacci representations of and in parallel, where is the Tribonacci constant, and $x= \lfloor n ψ\rfloo…
On a sequence of Kimberling and its relationship to the Tribonacci word
Lubomíra Dvořáková, Edita Pelantová, Jeffrey Shallit
In 2017, Clark Kimberling defined an interesting sequence of 's and 's by certain inflation rules, and he made a number of conjectures about thi…
Balanced Fibonacci word rectangles, and beyond
Jeffrey Shallit, Ingrid Vukusic
Following a recent paper of Anselmo et al., we consider rectangular matrices formed from the Fibonacci word, and we show that their balance properties can be solved wi…
New properties of the -representation of integers
Jeffrey Shallit, Ingrid Vukusic
We prove a few new properties of the -representation of integers, where . In particular, we prove a 2012 conjecture of Kimberling. As software assistants, we…
On the -adic valuations of values of Legendre polynomials
Max A. Alekseyev, Tewodros Amdeberhan, Jeffrey Shallit +1
We prove an explicit formula for the -adic valuation of the Legendre polynomials evaluated at a prime , and generalize an old conjecture of the third author. We also…