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PYTHAGOREAN TRIPLES A. F. HORADAM University of New England, Armidale, (Submitted June 1980) Australia Define the sequence {wn} byCD w0 = a, w1 - b, wn+2 = wn + 1 +wn (n integer >_ 0; a, b real and both not zero). Then [2], (2) <» n *>n + 3> 2 + <2Wn+A+2>2 Freitag [1] asks us to find a cn9 (3) <^„+3' W = <*>» + ! + »+2>2- if it exists, for which 2F n+lFn+2> °n) is a Pythagorean triple, where Fn is the nth Fibonacci number. show that on = F2n+3. Earlier, Wulczyn [3] had shown that (4) (LnLn+39 2Ln+1Ln+2, It is easy to 5F2n+3) is a Pythagorean triple, where L n is the nth Lucas number. Clearly, (3) and (4) are special cases of (2) in which a = 0, b = 1, and a = 2, 2? = 1, respectively. One would like to know whether (3) and (4) provide the only solutions of (2) in which the third element of the triple is a single term. Our feeling is that they do. Now (5) W SO n + = aFn-l 1 bFn, z ' {a + b )F2n+3 (6) + {lab - {b2 + 2ab)F2n+3 b2?2n+s - 0 if a = 2b (7) a + (a2 - 2ab)F2n+2 • if a tt2F2n+3 a2)Fln+2 if b = 0 F 2n + 1 5a2F2n+1 if b = -2a II whence ' bFn (8) wn — or bLn or -aLn-i •< oFn_x results which may be verif ied in (5). 121 1 I by 122 COMPOSITION ARRAYS GENERATED BY FIBONACCI NUMBERS [May Therefore, only the Fibonacci and Lucas sequences, and (real) multiples of them, satisfy our requirement that the right-hand side of (2) reduce to a single term. References 1. 2. 3. H. Freitag. Problem B-426. The Fibonacci Quarterly 18, no. 2 (1980):186. A. F. Horadam. "Special Properties of the Sequence wn(a, b; p, q)." The Fibonacci Quarterly 5, no. 5 (1967):424-434. G. Wulczyn. Problem B-402. The Fibonacci Quarterly 18, no. 2 (1980):188. ***** COMPOSITION ARRAYS GENERATED BY FIBONACCI NUMBERS V. E. HOGGATT, JR. San Jose (Deceased) and MARJORIE BICKNELL-JOHNSON State University, San Jose, (Submitted September 1980) CA 95192 The number of compositions of an integer n in terms of ones and twos [1] is Fn+1, the (n + l)st Fibonacci number, defined by F0 = 0 , F1 = 1, and Fn42 = Fn+l +Fn . : Further, the Fibonacci numbers can be used to generate such composition arrays [2], leading to the sequences A = {an} and 5 = {bn}9 where (an9 bn) is a safe pair in Wythoff's game [3], [4], [6]. We generalize to the Tribonacci numbers Tn9 where T0 = 0, Tx = T2 = 1, and Tn+3 = Tn+Z + Tn+1 + Tn. The Tribonacci numbers give the number of compositions of n in terms of ones, twos, and threes [5], and when Tribonacci numbers are used to generate a composition array, we find that the sequences A - {An}, B = {Bn}9 and C = {Cn} arise, where An9 Bn9 and C n are the sequences studied in [7]. 1. The Fibonacci Composition Array To form the Fibonacci composition array, we use the difference of the subscripts of Fibonacci numbers to obtain a listing of the compositions of n in terms of ones and twos, by using Fn^1, in the rightmost column, and taking the Fibonacci numbers as placeholders. We index each composition in the order in which it was written in the array by assigning each to a natural number taken in order and, further, assign the index k to set A if the kth composition has a one in the first position, and to set B if the kth composition has a two in the first position. We illustrate for n - 6,using F7 to write the rightmost