21st Annual Student Research Conference
Self-Generating Sets and Numeration Systems
Joseph A. Palmer
Dr. David Garth, Faculty Mentor
I will explore connections between self-generating sets and numeration systems. I define a self-generating set S generated by a set of functions, F. Kimberling noticed that the set generated by F={2x, 4x + 1} is the set of natural numbers whose binary expansions contain no consecutive ones. These expansions correspond to the greedy expansions of the natural numbers with respect to the Fibonacci sequence. I consider the question of which numeration systems of the natural numbers contain digit expansions that can be realized as the base two expansions of the integers in a self-generated set. I give conditions for a numeration system over {0, 1} to have a base sequence. This will be used to generate several examples of based numeration systems that are self-generated. I also show that the base sequence of any self-generated numeration system must satisfy a linear recurrence relation.
Keywords: self-generating set, numeration system, binary expansion, Fibonacci sequence, base sequence, linear recurrence relation
Topic(s):Mathematics
Presentation Type: Oral Paper
Session: 52-1
Location: VH 1428
Time: 2:45