UNIQUENESS OF REPRESENTATIONS BY MORGAN-VOYCE NUMBERS. A. F. Horadam The University of New England, Armidale, Australia 2351 (Submitted June 1998)

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1 with Consider the recurrence UNIQUENESS OF REPRESENTATIONS BY MORGAN-VOYCE NUMBERS A. F. Horadam The University of New England, Armidale, Australia 5 (Submitted June 99). MORGAN-VOYCE NUMBERS X n + = ^,, + - % n (-) X 0 = a, X x = b (a, * integers). (.) Morgan-Voyce numbers B n, h n, and their related numbers C n9 c n are then generated according to the following scheme in which F n9 L n symbolize the rfi Fibonacci and 7 th Lucas numbers, respectively: (B) (*) ( Q (c) x B c c n a ~6~ ~T - Readers are encouraged to determine the first few members of each of these sequences. In particular, {BJ = 0,,,,,55,... The sets of numbers (.) are special cases of the corresponding sets of polynomials B n (x), h (x),c n (x),c n (x)[]whenx = L b II " w ^m ~, Pln-l L n ^n-\ (.). REPRESENTATIONS BY B m Next, consider the representation of positive integers N by means of B n : # = > / f l (^=0,,). (.) /=i Of special interest is the case as in [] in which all the a t in (.) are, giving rise to the numbers,,,,..., i.e.,! = i W - l. (-) /= A minimal representation is indicated in the abbreviated table (Table ) in which an empty space signifies 0 (zero). This table has already appeared in []. An essential feature of this representation proved in [] is that no two successive terms in the summation have coefficient. [JUNE-JULY

2 AT TABLE. Minima! Representation for {BJ: n =,,, "T l B > ii_ Is this representation B unique? N fe t ~T ~T # # ~~ to L B ~~r Write S for the set of digits 0,, of length in the representation. Let /jt " = the smallest integer in S y j-m&x _ ^ e i a r g e s t integer in 5^, I i ^ = the range of integers in S, \I = the number of integers in S. Then we readily construct the following scheme (Table ). TABLE * B n Representation Summary 5 s s S, s S s Rt U,...,7,...,0,...,5 Nf B, B> B B N? * B - l 5 - l 5, - B 5 -l '* = F 5 = F 5 = F 7 = F 9 Jfc s Fit,...,F M+ -l B t = F it B M ~ = ^*+ _ F l+\ Clearly, I = JVp» - # P + = ^ - - F = F, +. In each bloc of length in Table, ( the smallest number is necessarily (0, 0, 0,..., ), and the largest number is necessarily (0, 0, 0,..., ). Lemma : B n <N<B n+l -l. E.g., Bs(= 97)<N = 000<5^-(= 5). 000]

3 Lemma : is uniquely determined by N. E.g., N = \000=> = 5. Combining the above information, we deduce that Theorem : Every positive integer N has a unique representation of the form where [] two successive values a h a i+l cannot both be. The distinctive pattern fixed in Tables and determines the uniqueness of the representation. A tabular schedule similar to that in Table (but suppressed here for the sae of brevity) ought now to be constructed for maximal representations by B. The embargo on the appearance of two successive coefficients in the summation with the value, as in the enunciation of Theorem, naturally does not apply for maximality. A fixed pattern of the coefficients emerges in the tabulation of maximal representations for B m leading to the conviction that the maximal representation is unique. Where this situation differs from that, say, for Pell numbers [], is that, while (.) in which all coefficients are is there common to both minimal and maximal representations, other summations here are common to both which do not belong to (.), e.g., 5 = B l +B. Also see [] in this context. (i) C H (lacunary). OTHER REPRESENTATIONS Coming now to the companion number set {CJ =,,7,,7,... to {B }, i.e., (.)(C), we find that the even tenor of our progress is disrupted. For a start, C 0 =, Q =, so that there is no possible representation of (unity). Thus, any representation is necessarily lacunary. It is no good appealing to C j as an accommodating adjunct to the set {C n } since C_ t = (indeed, C_ = C n ). Because of this hiatus, there is also no member in the pattern of the minimal representation of, say, though it can be represented maximally as = C 0 +Q, in which there occur two successive coefficients equal to. Except for the lacuna at N = l, the potentially fixed minimal pattern is negated in a regular way at C n =, n >. The nature of the representation is therefore hybrid. (ii) Turning now to the Morgan-Voyce numbers {b n }:,,5,,,..., we encounter a similar set of circumstances to those for {BJ. Arguments paralleling those employed in the previous section are liewise applicable to this context. Analogously to Table, a minimal representation table may be constructed (an entertaining and instructive pastime). As for B n, the proscription of two successive coefficients equal to in a minimal representation applies here also. For comparison with the Table Summary for JJ n,-we here append a Summary (Table ) for h n, in which non-capital symbols correspond to the capital symbols specified in (.). [JUNE-JULY

4 TABLE * b u Representation Summary % s i S S S h F h,..., 5,...,,..., -i?» Mjt+i ~~ ^r" *. b i 6* max b -i * " * " l * * + -! Observe that, by (.), i = (^+ -) - (^) + = b +l -b = F +l - F^ = F. Uniqueness of the minimal representation is determined by the fixedness of the pattern. (iii) c n Some initial comfort is offered here by the fact that = c? = c l. But to represent the number, we need to revert to the subterfuge of including - = c_ x (c_ n = c n in fact) in our set {c n }. This implies that a representation exists which is non-lacunary. There is a purposefulness about the coefficients which then suggests minimality and uniqueness. '* ^ ^ ^ ^ ^ * Write. CONCLUDING OBSERVATIONS n-l & =,. (.), b = 5>, %, = XC,, c = c,.. = J=l j = 0 J=l Then we discover the following schedule (cf. (.)): m n ^n c Fibonacci Equivalence F n+l~ l F n ^n+l ~~ ^w~ Recurrence Relation ^ = ^ - * - += +\- h n %H- = % i + l " ~ % i _ C rc+ = C «+~~ C H + Aspects of S W and % n are discussed in [], while features of b n and c n are analyzed in []. Peripherally of import to this paper, but also to provide some publicity for the concept, we mention Brahmagupta polynomials [5] which relate to B n (x) and b n (x) [5], and to C (x) and c n ( x ) Nl- Historical information on Brahmagupta and his mathematics is given in some detail in [6]. REFERENCES. A. F. Horadam.!f Minmax Sequences for Pell Numbers." In Applications of Fibonacci Numbers 6;-9. Ed. G. E. Bergum, A. N. Philippou, & A. F. Horadam. Dordrecht: luwer, A. F. Horadam. "New Aspects of Morgan-Voyce Polynomials. In Applications of Fibonacci Numbers 7:6-76. Ed. G. E. Bergum, A. N. Philippou, & A. F. Horadam. Dordrecht: luwer, ] 5

5 . A. F. Horadam. "Unit Coefficients Sums for Certain Morgan-Voyce Numbers." Notes on Number Theory and Discrete Mathematics. (997): A. F. Horadam.. "Representation Grids for Certain Morgan-Voyce Numbers." The Fibonacci Quarterly A (999): E. R. Suryanarayan. "The Brahmagupta Polynomials." The Fibonacci Quarterly, (996): A. Weil. Number Theory: An Approach Through History: From Hammurapi to Legendre. Boston: Birhauser, 9. AMS Classification Number: B7 Announcement NINTH INTERNATIONAL CONFERENCE ON FIBONACCI NUMBERS AND THEIR APPLICATIONS July 7-JuSy, 000 L O C A L C O M M I T T E E Institut Stiperieur de Tecfaeologie Grand Duche de Luxembourg INTERNATIONAL C O M M I T T E E J. Lahr, Chairman A. F. Horadam (Australia), Co-chair M. Johnson (USA.) R. Andre-Jeannin A. N. Philippou (Cyprus), Co-chair P. iss (Hungary) M. Malvetti C. Cooper (USA.) G. M. Phillips (Scotland) C. Molitor-Braun P. Filipponi (Italy) J. Turner (New Zealand) M. Oberweis H. Harborth (Germany) M. E. Waddill (US A.) P. Schroeder Y. Horibe (Japan) L O C A L I N F O R M A T I O N For information on local housing, food, tours, etc., please contact: PROFESSOR JOSEPH LAHR Institut Superior de Technologie 6, rue R. Coudenhove-alergi L-59 Luxembourg joseph.lahr@ist.lu Fax: (005) Phone: (005) 00- C A L L F O R PAPERS Papers on all branches of mathematics and science related to the Fibonacci numbers, number theoretic facts as well as recurrences and their generalizations are welcome. Abstracts, which should be sent in duplicate to F. T. Howard at the address below, are due by June, 000. An abstract should be at most one page in length (preferably half a page) and should contain the author's name and address. New results are especially desirable; however, abstracts on wor in progress or results already accepted for publication will be considered. Manuscripts should not be submitted. Questions about the conference should be directed to: PROFESSOR F. T. HOWARD Wae Forest University Box 7 Reynolda Station Winston-Salem, NC 709 (U.S.A.) howard@mthcsc.wfu.edu 6 [JUNE-JULY