Generalizations of the Dual Zeckendorf Integer Representation Theorems—discovery by Fibonacci Trees and Word Patterns

In this paper we show how the two well-known integer representation theorems which are associated with the name of Zeckendorf may be generalized as dual systems by constructing colored tree sequences whose shade sets partition Z = {1, 2, . . . } . Many interesting properties of the representations can be observed directly from the tree diagrams, and the proofs of the properties can truly be said to be "evident" or "obvious"; we shall not translate such proofs into other symbolic forms. The Zeckendorf theorems are about representations of positive integers as sums of distinct elements of given number sequences. The first theorem is in Lekkerkerker [6], and a dual of it is given by Brown [2]. Early papers on properties of the Zeckendorf integer representations are Zeckendorf [12] and Brown [1]. Klarner [5] gives an excellent review of the literature to 1966, and extends many of the theories to that date. In [3] Carlitz et al. (1972) define Fibonacci representations of integers, and study their properties. In Turner [7] we showed how to construct certain tree sequences and defined their shade sets, which together demonstrated the Zeckendorf representation theorems. In Zulauf & Turner [13], we showed how the shade sets could be defined in a set-theoretic notation, and proved the Zeckendorf theorems in a concise manner. In Turner [8] and Turner & Shannon [9] we defined Fibonacci word patterns and used them to study tree shade sets.