 
Summary: EVERY POSITIVE KBONACCILIKE SEQUENCE EVENTUALLY
AGREES WITH A ROW OF THE KZECKENDORF ARRAY
PETER G. ANDERSON AND CURTIS COOPER
Abstract. For k 2, a fixed integer, we work with the kbonacci 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 kbonacci recurrence eventually agrees with
a row of the kZeckendorf array.
Throughout this paper, k 2 is a fixed integer.
Definition 1. The kbonacci sequence {Xn} is given by the recurrence
Xn = 0 for k + 2 n 0,
X1 = 1,
Xn =
k
i=1
Xni 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 wellknown theorem [6] (see also [4]).
(Strictly speaking, Zeckendorf's Theorem applies to the Fibonacci numbers (k = 2), but the
