2008 Student Research Conference:

21st Annual Student Research Conference

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