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PATTERNS IN DIFFERENCES BETWEEN ROWS IN k-ZECKENDORF LARRY ERICKSEN AND PETER G. ANDERSON
 

Summary: PATTERNS IN DIFFERENCES BETWEEN ROWS IN k-ZECKENDORF
ARRAYS
LARRY ERICKSEN AND PETER G. ANDERSON
Abstract. For a fixed integer k 2, we study the k-Zeckendorf array, Xk, based upon the
k-th order recurrence un = un-1 + un-k. We prove that the pattern of differences between
successive rows is a k-letter infinite word generalizing the infinite Fibonacci.
1. Introduction and Background
Definition 1. The k-Zeckendorf array [3], Xk = {xr,c | r, c 0}, is a doubly subscripted
array of positive integers. The first row begins with x0,c = c + 1 for 0 c < k. For i k,
x0,i = x0,i-1 + x0,i-k. (1)
Subsequent rows are specified inductively as follows. For r > 0, xr,0 is the smallest integer
not in previous rows. Let the k-Zeckendorf representation (see Definition 2 below) of xr,0 be
m
i=0 dix0,i. Then for c > 0,
xr,c =
m
i=0
dix0,i+c. (2)
Definition 2. The k-Zeckendorf representation of n is m
i=0 dix0,i, where for all i, di {0, 1}

  

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

 

Collections: Computer Technologies and Information Sciences