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 Collections: Computer Technologies and Information Sciences