IJPAM: Volume 10, No. 1 (2004)
OF INTEGERS: A 3-SHIFT THEOREM
Department of Computer Science and Applied Mathematics
Weizmann Institute of Science
Rehovot 76100, ISRAEL
Abstract.Let be two irrational numbers satisfying . Then the sequences , , , are complementary over , thus satisfies: , (, the smallest positive integer not in the set ). Suppose that is an integer. Then for all .
We define the following generalization of sequences , : Let , and let be an arbitrary finite set. Let , , . Let . We show that no matter how we pick and , from some point on the shift sequence assumes either one constant value or three successive values; and if the second case holds, it assumes these values in a very distinct fractal-like pattern, which we describe.
This work was motivated by a generalization of Wythoff game to
Received: September 11, 2003
AMS Subject Classification: 05A17, 91A05
Key Words and Phrases: complementary sequences, integer part function, 3-shift theorem, Wythoff games
Source: International Journal of Pure and Applied Mathematics