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} Collections: Computer Technologies and Information Sciences