rC THE OFFICIAL JOURNAL OF THE FIBONACCI ASSOCIATION TABLE OF CONTENTS Elliptic Functions and Lambert Series in the Summation of Reciprocals in Certain Recurrence-Generated Sequences. A.F. Horadam 98 A Matrix Approach to Certain Identities Piero Filipponi & A.F. Horadam 115 On the Number of Solutions of the Diophantine Equation ( J = ( \ \ - • Peter Kiss 111 Third International Conference Announcement 130 De Moivre-Type Identities for the Tribonacci Numbers Pin-Yen Lin 131 Entropy of Terminal Distributions and the Fibonacci Trees Yasuichi Horibe 135 The Length of a Three-Number Game Joseph W. Creely 141 Some Identities for Tribonacci Sequences S. Pethe 144 Fibonacci Sequence of Sets and Their Duals ..A. Zulauf& J.C. Turner 152 Second International Conference Proceedings 156 On the IP -Discrepancy of Certain Sequences .. Z,. Kuipers & J. -S. Shiue 157 A Partial Asymptotic Formula for the Niven Numbers Curtis N. Cooper & Robert E. Kennedy 163 On Certain Divisibility Sequences ..R.B. McNeill 169 Integral 4 by 4 Skew Circulants William C. Waterhouse 111 On A Result Involving Iterated Exponentiation R.M. Sternheimer 178 Elementary Problems and Solutions Edited by A.R Hillman 181 Advanced Problems and Solutions 9 8 * Edited by Raymond E* Whitney 186 VO]L UME 26 MAY 1988 NUMBER 2 PURPOSE The primary function 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, challenging problems, and innovative proofs of old ideas. EDITORIAL POLICY THE FIBONACCI 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 new ideas that develop enthusiasm for number sequences or the explora- tion of number facts. Illustrations and tables should be wisely used to clarify the ideas of the manuscript. 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THE FIBO- NACCI 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. Reprints can also be purchased from UMI CLEARING HOUSE at the same address. 1988 by © The Fibonacci Association All rights reserved, including rights to this journal issue as a whole and, except where otherwise noted, rights to each individual contribution. Hfe Fibonacci Quarterly Founded In 1963 by Verner E. Hoggatt, Jr. (1921-1980) and Br. Alfred Brousseau THE OFFICIAL JOURNAL OF THE FIBONACCI ASSOCIATION DEVOTED TO THE STUDY OF INTEGERS WITH SPECIAL PROPERTIES EDITOR GERALD E. BERGUM, South Dakota State University, Brookings, SD 57007-0194 ASSISTANT EDITORS MAXEY BROOKE, Sweeny, TX 77480 JOHN BURKE, Gonzaga University, Spokane, WA 99258 PAUL F. BYRD, San Jose State University, San Jose, CA 95192 LEONARD CARLITZ, Duke University, Durham, NC 27706 HENRY W. GOULD, West Virginia University, Morgantown, WV 26506 A.P. HILLMAN, University of New Mexico, Albuquerque, NM 87131 A.F. HORADAM, University of New England, Armidale, N.S.W. 2351, Australia FRED T. HOWARD, Wake Forest University, Winston-Salem, NC 27109 DAVID A. KLARNER, University of Nebraska, Lincoln, NE 68588 RICHARD MOLLIN, University of Calgary, Calgary T2N 1N4, Alberta, Canada JOHN RABUNG, Randolph-Macon College, Ashland, VA 23005 DONALD W. ROBINSON, Brlgham Young University, Provo, UT 84602 LAWRENCE SOMER, Catholic University of America, Washington, D.C. 20064 M.N.S, SWAMY, Concordia University, Montreal H3C 1M8, Quebec, Canada D.E. THORO, San Jose State University, San Jose, CA 95192 CHARLES R. WALL, Trident Technical College, Charleston, SC 29411 WILLIAM WEBB, Washington State University, Pullman, WA 99163 BOARD OF DIRECTORS OF THE FIBONACCI ASSOCIATION CALVIN LONG (President) Washington State University, Pullman, WA 99163 G.L. ALEXANDERSON Santa Clara University, Santa Clara, CA 95053 PETER HAGIS, JR. Temple University, Philadelphia, PA 19122 RODNEY HANSEN Whitworth College, Spokane, WA 99251 MARJORIE JOHNSON (Secretary-Treasurer) Santa Clara Unified School District, Santa Clara, CA 95051 JEFF LAGARIAS Bell Laboratories, Murray Hill, NJ 07974 LESTER LANGE San Jose State University, San Jose, CA 95192 THERESA VAUGHAN University of North Carolina, Greensboro, NC 27412 ELLIPTIC FUNCTIONS AND LAMBERT SERIES IN THE SUMMATION OF RECIPROCALS IN CERTAIN RECURRENCE-GENERATED SEQUENCES A. F. HORADAM University of New England, Armidale, Australia (Submitted July 1986) 1. INTRODUCTION Consider the sequence of positive integers {wn} defined by the recurrence relation Wn+2 = PWn + l - Wn C1'1) with initial conditions wQ = a, w1 - b, (1.2) where a ^ 0, b > 1, p > 1, q ^ 0 are integers with p 2 ^ kq. We first consider the "nondegenerate" case: p 2 > 4q. Roots of the characteristic equations of (1.1), namely, X2 - pA + q = 0 (1.3) are (a = (p + /p2 - 4q)/2, (1.4) (B = (p - /p2 - 4(7) /2. Note a > 0, B < 0 depending on q ^ 0. Then 2 a + 6 = p, aB = q, a - B = Vp - 4<? > 0. (1.5) The explicit Binet form for u n is w .A*l^|£ (1.6) in which = fe - a B s (1.7) ft: 1) - aa. It is the purpose of this paper to investigate the infinite sums 00 -I L 7T (1-8) n = l wn 00 -I E 7T- (1-9) E ^ 1 — • d.io) n = l w 2n-l 98 [May ELLIPTIC FUNCTIONS AND LAMBERT SERIES Special cases of {wn} which interest us here are: the Fibonacci sequence {Fn}i a = 0 9 b = l 9 p = l s q = - l ; (1.11) the Lucas sequence {Ln}i a = 2, b = 1, p = 1, q = -1; (1.12) the Fell sequence {Pn}: a = Q, 2 ? = l , p = 2, q = -l; (1.13) the Fell-Lucas sequence {Qn}: a = 2 9 b = 2 , p=2, q=-l; (1.14) the Fermat sequence {fn}: a = 0S b = 1, p = 3, q = 2; (1.15) the "Fermat-Lucas" sequence ign}- a = 2, Z? = 3, p = 3, g = 2; (1.16) the generalised Fibonacci sequence {Un}: a = 0, b = 1; (1.17) the generalised Lucas sequence {Vn}: a = 25 b = p. (1.18) The Fermat sequence (1.15) is also known as the Mersenne sequence. Binet forms and related information are readily deduced for (.1.11)-(1.18) n n from (1.4)-(1.7). Notice that fn = 2 - 1, gn = 2 + 1, and, for both (1.15) and (1.16), a = 2, g = 1, in which case the roots of the characteristic equa- tion are not irrational. Sequences (1.11), (1.13), (1,15), and (1.17), in which a = 0, b = 1, may be alluded to as being of Fibonacci type. On the other hand, sequences (1.12), (1.14), (1.16), and (1.18), in which a = 2, b = p, may be said to be of Lucas type. For Fibonacci-type sequences, we have A = B - 1, and the Binet form (1.6) reduces to _. n r> n n a - g ' (1.6) ' whereas for Lucas-type sequences, in which A B = a - 35 we have the simpler form n (1.6)" wn = a + g\ From (1.6), W„ 4a" lim lim lim (1.19) n ->• oo _1_ w„ 4an ->n + l 4 - 5 (!)" lim w + 1 = — since < 1 '(f)' < 1 a > 1. To prove this last assertion, we note that 2a = p + Vp2 - kq > 1 + 1 = 2. If p + Vp2 - 4(7 = 2, then q = p - 1; but q ^ 0, s o p ^ l = ^ p > l = » a > l . 1988] 99 ELLIPTIC FUNCTIONS AND LAMBERT SERIES Thus, oo . E — converges absolutely. (1,20) n= 1 Wn All the sequences (l.ll)-(l.18) satisfy (1.20). 2. BACKGROUND H i stor ical The desire to evaluate t~ (2.1) n = i t n seems to have been stated first by Laisant [21] in 1899 in these words: "A-t-on deja etudie la serie 1 I I 1 I 1 1 2 3 5 •••' que fovrnent les inverses des termes de Fibonacci3 et qui est evidemment oonvergentel" Barriol [3] responded to this challenge by approximating (2.1) to 10 deci- mal places: L -~- = 3.3598856662..