 
Summary: MULTIDIMENSIONAL ZECKENDORF REPRESENTATIONS
PETER G. ANDERSON AND MARJORIE BICKNELLJOHNSON
Abstract. We generalize Zeckendorf's Theorem to represent points in Zk1
, uniquely, as
sums of elements of orderk linear recurrences.
1. Background and Definitions
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 . (1)
When k = 2, {Xn} is the Fibonacci sequence, when k = 3 the tribonacci sequence, and so
on. Our purpose herein is to generalize the following wellknown theorem [5] (see also [2, 3,
4] 1
.
Theorem 1. Zeckendorf's Theorem. Every nonnegative number, n, is a unique sum of
distinct kbonacci numbers:
