2010 Student Research Conference:
23rd Annual Student Research Conference

Numeration Systems and Fractal Sequences
Joseph A. Palmer
Dr. David Garth, Faculty Mentor

It is well-known that every natural number can be written uniquely as a sum of nonconsecutive Fibonacci numbers, and thus the Fibonacci numbers can be used as a nonstandard base sequence for the natural numbers. This Fibonacci numeration system is used to construct the famous Wythoff array. We generalize this idea to other sequences and examine the various properties of the numeration systems generated. For our purposes an abstract numeration system is considered to be a collection of strings of digits over a finite alphabet. We use these abstract numeration systems to construct an array similar to the Wythoff array and examine its properties. In his 1995 paper Kimberling considered the self-similarity properties of the paraphrase sequence of the Wythoff array. We expand Kimberling's results to the more general case, and derive conditions on the base sequence for the associated paraphrase to have the properties in question.

Keywords: Abstract Numeration System, Fractal , Wythoff Array, Paraphrase Sequence

Topic(s):Mathematics

Presentation Type: Oral Paper

Session: 17-3
Location: MG 1098
Time: 10:00

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