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^SSffln^l THE OFFICIAL JOURNAL OF THE FIBONACCI ASSOCIATION VOLUME 20 FEBRUARY NUMBER 1 1982 CONTENTS On the Use of Fibonacci Recurrence Relations in the Design of Long Wavelength Filters and Interferometers . L.G. Batten 1 On the Convergence of Iterated Exponentiation—III Michael Creutz & R.M. Sternheimer 7 Generalized Fermat and Mersenne Numbers Steve high & Pat Jones 12 Fibonacci Numbers of Graphs Helmut Prodinger & Robert F. Tichy 16 Some Identities and Divisibility Properties of Linear Second-Order Recursion Sequences .............. Neville Rohhins 21 Polygonal Products of Polygonal Numbers and the Pell Equation ...... L.C. Bggan, Peter C. Eggan & J.L. Self ridge 24 Waiting for the Kth Consecutive Success and The Fibonacci Sequence of Order K A.N. Philippou & A. A. Muwafi 28 The Length of the Four-Number Game ...... William A. Webb 33 Sums of Consecutive Intergers . Robert Guy 36 Concerning a Paper by L.G. Wilson A.G. Shannon & A.F. Horadam 38 A Generalization of the Dirichlet Product . K. Joseph Davis 41 Combinatorial Aspects of an Infinite Pattern of Integers .. A.F. Horadam & A.G. Shannon 44 One-Pile Time and Size Dependent Take-Away Games Jim Flanigan 51 Fibonacci-Cayley Numbers . P.V. Satyanarayana Murthy 59 Generalizations of Some Problems on Fibonacci Numbers . P.V. Satyanarayana Murthy 65 A Variant of the Fibonacci Polynomials Which Arises in the Gambler's Ruin Problem . Mark E. Kidwell '& Clifford M. Hurvich 66 Generalized Fibonacci Numbers by Matrix Methods .. Dan Kalman 73 Explicit Descriptions of Some Continued Fractions J.O. Shallit 77 The Nonexistence of Quasiperfect Numbers of Certain Forms ....... ..................... Graeme I, Cohen 81 Representations of Every Integer as the Difference of Powerful Numbers Wayne L. McDaniel 85 Elementary Problems and Solutions ....... Edited by A.P. Hillman 89 Advanced Problems and Solutions Edited by Raymond E. Whitney 93 Hfe Fibonacci Quarterly Founded in 1963 by Verner E. Hoggatt, Jr. (1921-1980), Br. Alfred Brousseau, and LD. Ruggles THE OFFICIAL JOURNAL OF THE FIBONACCI ASSOCIATION DEVOTED TO THE STUDY OF INTEGERS WITH SPECIAL PROPERTIES EDITOR Gerald E* Bergum BOARD OF DIRECTORS G.L. Alexanderson (President), Leonard Klosinski (Vice-President), Marjorie Johnson (Secretary), Dave Logothetti (Treasurer), Richard Vine (Subscription Manager), Hugh Edgar and Robert Giuli. ASSISTANT EDITORS Maxey Brooke, Paul F. Byrd, Leonard Carlitz, H.W. Gould, A.P. Hillman, A.F. Horadam, David A. Klarner, Calvin T. Long, D.W. Robinson, M.N.S. Swamy, D.E. Thoro, and Charles R. Wall. EDITORIAL POLICY The principal purpose of The Fibonacci Quarterly is to serve as a focal point for widespread interest in the Fibonacci and related numbers, especially with respect to new results, research proposals, and challenging problems. The Quarterly seeks articles that are intelligible yet stimulating to its readers, most of whom are university teachers and students. These articles should be lively and well motivated, with innova- tive ideas that develop enthusiasm for number sequences or the exploration of number facts. Articles should be submitted in the format of the current issues of the Quarterly. They should be typewritten or reproduced typewritten copies, double spaced with wide margins and on only one side of the paper. Articles should be no longer than twenty-five pages. The full name and address of the author must appear at the beginning of the paper directly under the title. Illustrations should be carefully drawn in India ink on separate sheets of bond paper or vellum, approximately twice the size they are to appear in the Quarterly. Authors who pay page charges will receive two free copies of the issue in which their article appears. Two copies of the manuscript should be submitted to GERALD E. BERGUM, DEPARTMENT OF MATHEMATICS, SOUTH DAKOTA STATE UNIVERSITY, BROOKINGS, SD 57007. The author is encouraged to keep a copy for his own file as protection against loss. Address all subscription correspondence, including notification of address changes, to SUB- SCRIPTION MANAGER, THE FIBONACCI ASSOCIATION, UNIVERSITY OF SANTA CLARA, SANTA CLARA, CA 95053. Annual domestic Fibonacci Association membership dues, which include a subscription to The Fibonacci Quarterly, are $20 for Regular Membership, $28.for Sustaining Membership I, $44 for Sustaining Membership II, and $50 for Institutional Membership; foreign rates, which are based on international mailing rates, are somewhat higher than domestic rates (please write for details). The Quarterly is published each February, May, August and November. All back issues of The Fibonacci Quarterly are available in microfilm or hard copy format from UNIVERSITY MICROFILMS INTERNATIONAL, 300 North Zeeb Road, Dept P.R., ANN ARBOR, MI 48106. ON THE USE OF FIBONACCI RECURRENCE RELATIONS IN THE DESIGN OF LONG WAVELENGTH FILTERS AND INTERFEROMETERS L. C. BOTTEN The New South Wales Institute of Technology P.O. Box 123, Broadway, N.S.W., 2007, Australia (Submitted May 1979) 1. INTRODUCTION The frequent occurrence of Fibonacci related numbers in art and nature has been of interest to readers of this journal since the very first issue when Basin [1] wrote such an article. The purpose of this paper is to explain briefly the use of Horadam*s general- ized Fibonacci recurrence relation [2] (1.1) un = pun_l - qun_2 in _> 2) in a problem in optics. If we consider the sequence {un} whose elements satisfy (1.1) with initial conditions u0 = 1 and u± = 5 0 , thens from Horadam, we have that n n un = Va + W$ 3 - S0 S. - a where V = ~TT - a 9» W 3 - a s in which a9 3 are the roots of the characteristic equation x2 - px + q - 0. 2. NONLINEAR RECURRENCE RELATION To show the relationship with the optics problems we need some more prelimi- nary results. Put p - B - A9 q = C - AB, and the recurrence relation can be rewritten as -2-i- - {B - A)—?- + (C - AB) = 0. un-l un-l We then add and subtract the term (Z*± + A) to get \ Un ) U U U U U n + 1 n n + 1 o n+i o n 2 ^Jl±l. + A—IL. + A-Jl£L + Az„ , 4 - ^ t i - - A - B —-AB + C = 09 which can be rewritten as From this we obtain the non-linear recurrence relation (2.1) R R n - Aff - BR . + C = 0, v ' n n-l n n-1 7 in which 1 ON THE USE OF FIBONACCI RECURRENCE RELATIONS IN THE DESIGN 2 [Feb. OF LONG WAVELENGTH FILTERS AND INTERFEROMETERS (2.2) ^ . I k t L + A. Thus, (2 3) R = U + a)7an + q + &)W$n Van .+ A/3^ where the term 5 0 in the definitions of 7 and ftf is given by S0 =i?0 - A. 3. THE OPTICS PROBLEM 3.1 General Remarks In this section, we will show that Eq. (2.1) occurs in the theory of multi- element optical filters and interferometers. However, before launching into its derivation, it is necessary to acquaint the reader with the background to the problem. Devices such as beam splitters, filters, and interferometers are common tools in all forms of experimental optics. In the fields of infrared physics and micro- wave engineering, the construction of such apparatus relies upon the use of wire meshes (or grids) [3]. It is possible to classify these structures into two dis- tinct classes according to their spectral properties. These are: a. inductive grids, made by perforating (in doubly-periodic fashion) a thin metal plate with aperatures, and b. capacitive grids, the natural complement of inductive grids, which are composed of a periodic array of metal inclusions immersed in an insulating material. The transmittance of inductive structures approaches zero at long wavelengths, whereas that of capacitive grids approaches unity at those wavelengths. In the far infrared and microwave regions of the electromagnetic spectrum, absorption within the metal is negligible and so we need only concern ourselves with the re- flectance and transmittance of these structures. With these prefatory remarks, let us now concern ourselves with the design of interferometers and low-frequency pass filters. These consist of a stack of many such grids separated from one another by a distance of s. In the case of the in- terferometer, the stack is composed of purely inductive elements, while the low- pass filter is composed of a stack of captive structures. Each of the grids in the stack acts as a diffraction grating and gives rise to an infinite set of diffracted plane waves (orders) excited by the incident plane wave field. Interferometers and filters are operated with wavelengths in excess of the grid period (d), and so it may be deduced that only the single undispersed wave is propagating (i.e., capable of carrying energy away from the grids). All of the other orders are said to be evanescent and decay exponentially as they propagate away from a grid. Provided that the ratio s/d exceeds 0.5, the evanes- cent orders provide no significant mechanism for communication between the grids [4] and so we need only consider the zeroth (or undispersed) order in our deriva- tion. 3.2 Derivation of a Non-Linear Difference Equation Let i?0 and T0 be the amplitude reflection and transmission coefficients of the zeroth order of one of these grids. Now consider a wave incident upon the (n+1)- grid structure of Figure 1. Let Rn and Tn be the amplitude reflection and trans- mission coefficients of this device relative to a phase origin at the center of the uppermost grid. We now regard this (n + l)-grid structure as a single grid displaced by a distance s from an n-grid structure. By adopting a multiple scat- 1982] ON THE USE OF FIBONACCI RECURRENCE RELATIONS IN THE DESIGN OF LONG WAVELENGTH FILTERS AND INTERFEROMETERS 3 tering approach, we can now trace the path of a wave through this system.
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    Repdigits in Euler Functions of Associated Pell Numbers

    Proc. Indian Acad. Sci. (Math. Sci.) (2020) 130:25 https://doi.org/10.1007/s12044-019-0551-3 Repdigits in Euler functions of associated pell numbers G K PANDA and M K SAHUKAR∗ Department of Mathematics, National Institute of Technology, Rourkela 769 008, India *Corresponding author. E-mail: [email protected]; [email protected] MS received 28 January 2019; revised 16 September 2019; accepted 22 October 2019 Abstract. In this paper, we discuss some properties of Euler totient function of associated Pell numbers which are repdigits in base 10. Keywords. Pell numbers; associated Pell numbers; Euler totient function; repdigits. Mathematics Subject Classification. Primary: 11A25 Secondary: 11B39. 1. Introduction The Euler totient function φ(n) of a positive integer n is the number of positive integers less than or equal to n and relatively prime to n.Ifn has the canonical decomposition = a1 ··· ar , ,..., , ,..., n p1 pr , where p1 p2 pr are distinct primes and a1 a2 ar are positive integers, then it is well-known that − − φ( ) = a1 1( − ) ··· ar 1( − ). n p1 p1 1 pr pr 1 The Pell sequence {Pn} is defined by means of a binary recurrence relation Pn+1 = 2Pn + Pn−1 with initial terms P0 = 0, P1 = 1. The associated Pell sequence {Qn} satisfies an identical recurrence relation with initial terms Q0 = 1, Q1 = 1. The sequence {Rn}, where Rn = 2Qn, is known as the Pell–Lucas sequence. The Binet formulas for Pell and associated Pell numbers are αn − βn αn + βn Pn = and Qn = , α − β 2 √ √ where α = 1 + 2 and β = 1 − 2.
  • A Note on Generalized K-Pell Numbers and Their Determinantal Representation -.:: Natural Sciences Publishing

    A Note on Generalized K-Pell Numbers and Their Determinantal Representation -.:: Natural Sciences Publishing

    J. Ana. Num. Theor. 4, No. 2, 153-158 (2016) 153 Journal of Analysis & Number Theory An International Journal http://dx.doi.org/10.18576/jant/040211 A Note on Generalized k-Pell Numbers and Their Determinantal Representation Ahmet Oteles¸¨ 1,∗ and Mehmet Akbulak2 1 Department of Mathematics, Faculty of Education, Dicle University, TR-21280, Diyarbakır, Turkey 2 Department of Mathematics, Art and Science Faculty, Siirt University, TR-56100, Siirt, Turkey Received: 3 Jan. 2016, Revised: 9 Apr. 2016, Accepted: 11 Apr. 2016 Published online: 1 Jul. 2016 Abstract: In this paper, we investigate permanents of an n × n (0,1,2)-matrix by contraction method. We show that the permanent of the matrix is equal to the generalized k-Pell numbers. Keywords: Pell sequence, Permanent, Contraction of a matrix 1 Introduction where the summation extends over all permutations σ of the symmetric group Sn. The well-known Pell sequence {Pn} is defined by the Let A = [ai j] be an m × n real matrix with row vectors recurrence relation, for n > 2 α1,α2,...,αm. We say A is contractible on column (resp. P = 2P + P row) k if column (resp. row) k contains exactly two n n−1 n−2 nonzero entries. Suppose A is contractible on column k where P1 = 1 and P2 = 2. with aik 6= 0 6= a jk and i 6= j. Then the (m − 1) × (n − 1) In [1], the authors defined k sequences of the matrix Ai j:k obtained from A by replacing row i with k α α generalized order-k Pell numbers Pn as shown: a jk i + aik j and deleting row j and column k is called the contraction of A on column k relative to rows i and j.