The Zeckendorf representation and the Golden Sequence

Transcription

1 University of Wollongong Researc Online Faculty of Engineering and Information Sciences - Papers: Part A Faculty of Engineering and Information Sciences 1991 Te Zeckendorf representation and te Golden Sequence Martin Bunder University of Wollongong, mbunder@uow.edu.au Keit Tognetti University of Wollongong, tognetti@uow.edu.au Publication Details Bunder, M. & Tognetti, K. (1991). Te Zeckendorf representation and te Golden Sequence. Te Fibonacci Quarterly: a journal devoted to te study of integers wit special properties, 29 (3), Researc Online is te open access institutional repository for te University of Wollongong. For furter information contact te UOW Library: researc-pubs@uow.edu.au

2 Te Zeckendorf representation and te Golden Sequence Abstract Te Zeckendorf representation of a number is simply te representation of tat number as te sum of distinct Fibonacci numbers. If te number of terms of tis sum is minimized, tat representation is unique, as also is te representation wen te number of terms is maximized. Keywords sequence, zeckendorf, golden, representation Disciplines Engineering Science and Tecnology Studies Publication Details Bunder, M. & Tognetti, K. (1991). Te Zeckendorf representation and te Golden Sequence. Te Fibonacci Quarterly: a journal devoted to te study of integers wit special properties, 29 (3), Tis journal article is available at Researc Online: ttp://ro.uow.edu.au/eispapers/1961

3 In wat follows, we ave Martin Bunder and Keit Tognetti Te University of Wollongong, N.S.W. 2500, Australia (Submitted August 1989) Preamble /S - 1 Te Golden section:, x = - = Fibonacci numbers: F Q = 0, F l = 1, F i = F i _ l + F i _ 2, i > 2. Te Zeckendorf representation of a number is simply te representation of tat number as te sum of distinct Fibonacci numbers. If te number of terms of tis sum is minimized, tat representation is unique, as also is te representation wen te number of terms is maximized. (See Brown [1] and [2].) A general Zeckendorf representation will be written as F k., were k x > k 2 > > > k > 2. J= 1 J Tus, 16 can be represented as F 7 + Fi+, F 6 + ^5 + * V F 7 + F 3 + F 2, and F e + F s + F 3 + F 2. Te first is te unique minimal representation; te last is te unique maximal representation. Te oters sow tat representations of any intermediate lengt need not be unique. It is easy to sow tat only numbers of te form F n - 1 ave a unique Zeckendorf representation (i.e., one tat is maximal and minimal). From ere on, we will refer to te minimal Zeckendorf representation and te maximal Zeckendorf representation as te miwimai and maximal. We define Beta-sequence: {3j} 5 J = 1, 2, 3,..., 3j = [ U + 1)T] - [ J T ]. Tis takes on only te values zero or unity. Golden sequence: Any sequence suc as abaababa... wic is obtained from te Beta-sequence $i» 32' $33» were "2?" corresponds to a zero and "a" corresponds to a unit. We will prove tat te final term of eac maximal representation is eiter F 2 or F 3 and sow te pattern associated wit te final terms in te representations of 1, 2, 3, 4, 5, 6,..., namely: F 2, F 3i F 2> F 2, F 3, F 2s... is a Golden sequence wit te term F 2 corresponding to a unit and te term ^3 corresponding to a zero. More specifically, we will sow tat te last term in te maximal representation of te number n is ^3-3 n = 2-3 n» We note a similar result for te "modified" Zeckendorf representation wic may include F± as well as F 2. Main Results Teorem 1: Te maximal ends wit F 2 or F 3. Proof: We note tat F 3 cannot be replaced by F 2 + F^ in a Zeckendorf expansion as F 2 = Fi. If F k wit k > 3 is te smallest term in an expansion of a number 1991] 217

4 n, ten F k can be replaced by ^ - i + F k^2 an & s o t n e expansion is not maximal. Tus, if an expansion is maximal, it must end in F^ or F3. Lemma 1: [ (j + ^ ) x ] = F i - l + [JT] if t > 2 and 0 < j < F i+l. Proof: Fraenkel, Muckin, and Tassa proved in [3] tat if 0 is irrational, 0 < «7 < tfi anc^ Vil^i i- s t n e ^ t convergent to 0 in te elementary teory of continued fractions, ten [(j + qi-i)q] = p i _ l + [j'0], > 1. As F^-i/Fi is a convergent to T, our result follows. Lemma 2: If ^T F k. is a Zeckendorf expansion, ten J] F k. < ^ n Proof: E \. * ^ ^ ^2 - ** 1 + i " 2, since ^ = ^n Te result is now obvious. Lemma 3: If j* as a Zeckendorf expansion XI ^ 5 t n e n i= 1 * (a) [ J T ] - F ki. x + ^ 2 _! F ^ _ i _ 1 + [ T ^ J (b) [ ( j + 1 ) T ] - F k! + F k l K! + F, Proof: (a) Let 7?z = X ^.» ten by Lemma 2, m < F^ +i and so by Lemma 1, i - 2 i [ J T ] - [ ( ^ + 772) T ] - F f c l _ l + [777T]. S i m i l a r l y, i f n = ^k ' t 7721 ] = F k 2 -l [JT] = ^ - l + * " * + V i " l + [ T F k J. (b) As i n (a) ( t i s time wit?77+ 1 < F^ +i), [U + D T ] - [{F ki (F k + 1 ) ) T ] + t n T ]> s o e v e n t u a l l y - F ki. x ^ _ x - l + l(f k + 1 ) T ] = F k l F^^-l + F k -l by Lemma 1. Lemma 4: If j as a maximal Y\ FT.., ten l i - 1 (a) [ J T ] = F f e i _! ^ _ x - i + ^ - 1. (b) B, F kfc. Proof: (a) If k = 2, ten UF k ] = 0 = F k - 1. Teorem If fc fc - 3, ten [TF k ] = 1 - F k - 1, so te r e s u l t follows by Lemma 3 ( a ). (b) By Lemmas 4(a) and 3 ( b ), Bj - [(J + D T ] - [ J T ] - F k. Y - F k + 1 = 2 - F fcfc, as ^ = 2 or 3 and so F k _ ]_ = 1. 2: Te l a s t term i n te maximal for j* i s ^3-3-- = [Aug.

5 Proof: By Lemma 4(b), if F k S 2 - F k. If k = 3, ten &j = 0 and F 3 $. is te last term in a maximal for j, ten = F 3 = 2 - Bj. If fe fe = 2 5 ten Bj = 1 and F 3 _ 3j = F 2 = 2 - Bj - We now see tat te last term of te maximal for any integer j is eiter 1 or 2. It also follows immediately tat te sequence of te last terms for te maximals for 1, 2, 3, 4,... form a Golden sequence , were a unit is uncanged but a zero is replaced by 2. Suppose we form te modified maximal from te maximal by forcing te last term to be unity; tat is, te last two terms are F3 + i^?' F 3 + Fi, or F^ + F\. Ten it follows easily from te above tat te second last terms of te modified maximals for 2, 3, 4,... correspond to te same golden pattern as te last terms in te maximals for 1, 2, 3,.... References 1. J. L. Brown. "Zeckendorf's Teorem and Some Applications." Fibonacci Quarterly 2.2 (1964): J. L. Brown. "Z New Caracterization of te Fibonacci Numbers." Fibonacci Quarterly 3.1 (1965): A. S. Fraenkel, M. Muskin, & U. Tassa. "Determination of [nq] by Its Sequence of Differences." Can. Mat. Bull. 21 (1978): Applications of Fibonacci Numbers Volume 3 New Publication Proceedings of 'Te Tird International Conference on Fibonacci Numbers and Teir Applications, Pisa? Italy, July 25-29, 1988.' edited by G.E. Bergtim, A.N. Pllippou and A.F. Horadam Tis volume contains a selection of papers presented at te Tird International Conference on Fibonacci Numbers and Teir Applications. Te topics covered include number patterns, linear recurrences and te application of te Fibonacci Numbers to probability, statistics, differential equations, cryptograpy, computer science and elementary number teory. Many of te papers included contain suggestions for oter avenues of researc. For tose interested in applications of number teory, statistics and probability, and numerical analysis in science and engineering. 1989, 392 pp. ISBN X Hardbound Dfl / 65.00/US $99.00 A.M.S. members are eligible for a 25% discount on tis volume providing tey order directly from te publiser. However, te bill must be prepaid by credit card, registered money order or ceck. A letter must also be enclosed saying " l a m a member of te American Matematical Society and am ordering te book for personal use." KLUWER ACADEMIC PUBLISHERS P.O. Box 322, 3300 AH Dordrect, Te Neterlands P.O. Box 358, Accord Station, Hingam, MA , U.S.A. 1991] 219