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MULTIDIMENSIONAL ZECKENDORF REPRESENTATIONS PETER G. ANDERSON AND MARJORIE BICKNELL-JOHNSON
 

Summary: MULTIDIMENSIONAL ZECKENDORF REPRESENTATIONS
PETER G. ANDERSON AND MARJORIE BICKNELL-JOHNSON
Abstract. We generalize Zeckendorf's Theorem to represent points in Zk-1
, uniquely, as
sums of elements of order-k linear recurrences.
1. Background and Definitions
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 . (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 well-known theorem [5] (see also [2, 3,
4] 1
.
Theorem 1. Zeckendorf's Theorem. Every nonnegative number, n, is a unique sum of
distinct k-bonacci numbers:

  

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

 

Collections: Computer Technologies and Information Sciences