2006 Student Research Conference:

19th Annual Student Research Conference

19th Annual Student Research Conference

**Self-Generating Sets, Missing Blocks, and Substitutions**

David M. Failing

Dr. David Garth, Faculty Mentor

We generalize the result of Allouche, Shallit, and Skordev regarding the Kimberling sequence, defined by: 1. 1 belongs to S; 2. if the integer x belongs to S, then 2x and 4x-1 are also in S; 3. nothing else is in S. Thus: S = {1, 2, 3, 4, 6, 7, 8, 11, 12, 14, 15,16,...} The set T=S-1, with its first term removed, is both the set of integers whose binary expansions contain no 00 block and the fixed point of the Fibonacci morphism 0->01, 1->0. We expand this result to the natural generalization of the Fibonacci morphism, of the form 0->0^n 1, 1->0.

**Keywords:** Kimberling sequence, self-generating sets, lazy expansion, missing blocks, Fibonacci sequence

**Topic(s):**Mathematics

**Presentation Type:** Oral Paper

**Session:** 51-2

**Location:** OP 2111

**Time:** 3:00