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EVERY POSITIVE K-BONACCI-LIKE SEQUENCE EVENTUALLY AGREES WITH A ROW OF THE K-ZECKENDORF ARRAY
 

Summary: EVERY POSITIVE K-BONACCI-LIKE SEQUENCE EVENTUALLY
AGREES WITH A ROW OF THE K-ZECKENDORF ARRAY
PETER G. ANDERSON AND CURTIS COOPER
Abstract. For k 2, a fixed integer, we work with the k-bonacci sequence, {Xn}, a kth
order generalization of the Fibonacci numbers, and their use in a Zeckendorf representation
of positive integers. We extend Zeckendorf representations using {Xn | n Z} and show that
every sequence of positive integers satisfying the k-bonacci recurrence eventually agrees with
a row of the k-Zeckendorf array.
Throughout this paper, k 2 is a fixed integer.
Definition 1. The k-bonacci sequence {Xn} is given by the recurrence
Xn = 0 for -k + 2 n 0,
X1 = 1,
Xn =
k
i=1
Xn-i for all n Z.
When k = 2, {Xn} is the Fibonacci sequence, when k = 3 the tribonacci sequence, and so
on. We have deliberately used Z as the domain of subscripts for {Xn}.
Our purpose herein is to generalize the following well-known theorem [6] (see also [4]).
(Strictly speaking, Zeckendorf's Theorem applies to the Fibonacci numbers (k = 2), but the

  

Source: Anderson, Peter G. - Department of Computer Science, Rochester Institute of Technology

 

Collections: Computer Technologies and Information Sciences