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TABLE of CONTENTS Elementary Problems and Solutions. .Edited By THE OFFICIAL JOURNAL OF THE FIBONACCI ASSOCIATION a TABLE OF CONTENTS On a New Kind of Numbers A.K. Agarwal 194 Some Sequences of Large Integers. Henry Ibstedt 200 Characteristics and the Three Gap Theorem .Tony van Ravenstein, Graham Winley and Keith Tognetti 204 Announcement on Fourth International Conference on Fibonacci Numbers and Their Applications 214 A Two-Variable Lagrange-Type Inversion Formula with Applications to Expansion and Convolution Identities Christian Krattenthaler 215 Lambert Series and the Summation of Reciprocals in Certain Fibonacci-Lucas-Type Sequences R. Andre- Jeannin 223 Third International Conference Proceedings 226 The Distribution of Residues of Two-Term Recurrence Sequences Eliot Jacobson 227 Generalizations of the Dual Zeckendorf Integer Representation Theorems—Discovery by Fibonacci Trees and Word Patterns J.C. Turner and T.D. Robb 230 The Fibonacci Sequence and the Time Complexity of Generating the Conway Polynomial and Related Topological Invariants Phillip G. Bradford 240 An Algebraic Indentity and Some Partial Convolutions W.C. Chu 252 On Certain Number-Theoretic Inequalities J. Sandor and L. Toth 255 Length of the /z-Number Game Anne Ludington-Young 259 Tiling the kth Power of a Power Series .-..../. Arkin, D.C. Arney, G.E. Bergum, S.A. Burr and B.J. Porter 266 On a Hoggatt-Bergum Paper with Totient Function Approach for Divisibility and Congruence Relations Sahib Singh 273 Elementary Problems and Solutions. .Edited by A.P.Hillman 277 Advanced Problems and Solutions Edited by Raymond E. Whitney 283 £ vVoOL UME 28 AUGUST 1990 NUMBER 3 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. Unanswered questions are encouraged, and a complete list of references is ab- solutely necessary. SUBMITTING AN ARTICLE Articles should be submitted in the format of the current issues of THE FIBONACCI QUARTERLY. They should be typewritten or reproduced typewritten copies, that are clearly readable, double spaced with wide margins and on only one side of the paper. The full name and address of the author must appear at the beginning of the paper directly under the title. Il- lustrations should be carefully drawn in India ink on separate sheets of bond paper or vellum, approximately twice the size they are to appear in print. Since the Fibonacci Association has adopted Fi = F2=l, F/i+/ = F/i + F/i-/Jn>2 and Li = l, L2 = 3, Ln+i = Ln + Ln-i, n > 2 as the stan- dard definitions for The Fibonacci and Lucas sequences, these definitions should not be a part of future papers. However, the notations must be used. Two copies of the manuscript should be submitted to: GERALD E. BERGUM, EDITOR, THE FIBONACCI QUARTERLY, DEPARTMENT OF COMPUTER SCIENCE, SOUTH DAKOTA STATE UNIVERSITY, BOX 2201, BROOKINGS, SD 57007-0194. Authors are encouraged to keep a copy of their manuscripts for their own files as protection against loss. The editor will give immediate acknowledgment of all manuscripts received. SUBSCRIPTIONS, ADDRESS CHANGE, AND REPRINT INFORMATION Address all subscription correspondence, including notification of address change, to: RICHARD VINE, SUBSCRIPTION MANAGER, THE FIBONACCI ASSOCIATION, SAN- TA CLARA UNIVERSITY, SANTA CLARA, CA 95053. Requests for reprint permission should be directed to the editor. However, general permission is granted to members of The Fibonacci Association for noncommercial reproduction of a limited quantity of individual articles (in whole or in part) provided complete reference is made to the source. Annual domestic Fibonacci Association membership dues, which include a subscription to THE FIBONACCI QUARTERLY, are $30 for Regular Membership, $40 for Sustaining Membership, and $67 for Institutional Membership; foreign rates, which are based on interna- tional mailing rates, are somewhat higher than domestic rates; please write for details. THE FIBONACCI 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. 1990 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. U35e Fibonacci Quarterly Founded in 1963 by Verner E. Hoggatt, Jr. (1921-1980) and Br. Alfred Brousseau (1907-1988) 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 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, Armidaie, N.S.W. 2351, Australia FRED T. HOWARD, Wake Forest University, Winston-Salem, NC 27109 CLARK KIMBERLING, University of Evansville, Evansville, IN 47722 DAVID A. KLARNER, University of Nebraska, Lincoln, NE 68588 RICHARD MOLLIN, University of Calgary, Calgary T2N 1N4, Alberta, Canada GARY L. MULLEN, The Pennsylvania State University, University Park, PA 16802 SUMIH OBAID, San Jose State University, San Jose, CA 95192 JOHN RABUNG, Randolph-Macon College, Ashland, VA 23005 DONALD W. ROBINSON, Brigham 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 99164-2930 BO AMD OF DIRECTORS THE FIBONACCI ASSOCIATION CALVIN LONG (President) Washington State University, Pullman, WA 99164-2930 G.L. ALEXANDERSON Santa Clara University, Santa Clara, CA 95053 PETER HAGIS, JR. Temple University, Philadelphia, PA 19122 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 ON A NEW KIND OF NUMBERS A. K. Agarwal The Pennsylvania State University, Mont Alto, PA 17237 (Submitted June 1988) 1. Introduction Recently, properties of the sequence {L2n+\}, where L2n+i is a Lucas number of order 2n + 1, were studied [1]. In the present paper, we introduce a new class of numbers defined by (i.D /(*, k) = ( - l r - ^ . Y ) ^ ! , where n is any nonnegative integer and 0 < k < n. These numbers have the interesting property (see [1], (1.5)): (1.2) £/(n, k) = 1, k= 0 for every nonnegative integral value of n. Property (1.2) is very much analo- gous to the following property of Stirling numbers of the first kind (see [2], (6), p. 145): (1-3) X>n = 0. k= 1 Also, these new numbers generalize the Catalan numbers in a nontrivial way. First, we recall that the Catalan numbers on are defined by means of the gen- erating relation ([5], p. 82) n 1 (1-4) £ cnt = (2t)' (l - /I - 4t), n= 0 or by the explicit formula ([5], p. 101) 1 (i.5) cn = (* + D ' ^ " ) . The following relationship is obvious: (-l)nf(n, 0) (1.6) on = (2n + 1) Results obtained in this paper include a table, recurrence relations, gen- erating functions, and summation formulas for these new numbers. In view of (1.6), many results reduce to their corresponding results for the Catalan num- bers found in the literature. In our last section, we pose two significant open problems. As usual (a)„ is PochhammerTs symbol and is defined by n . (1 if n = 0 K l} Ka)n \a(a + 1) ... (a + n - 1), for all n e {1, 2, 3, . } , 2&\ will denote the hypergeometric function defined by c 1 2 a-8> 2^i[ e. J - ^ - ^ r ^ r - *°' - ' ~ ' •••' and the Jacobi polynomials are defined by 194 [Aug. ON A NEW KIND OF NUMBERS (a 6) n '-n, -a - n; x + 1 (1.9) Pn ' M = n[ ("T-) 2^1 1 + 3; x - 1 2. Table of f(n, k) In this section, we give a table of f(n, k) produced by SCRATCHPAD—IBM's symbolic manipulation language. TABLE OF f(n, k) X 0 1 2 3 4 5 6 7 8 9 10 0 1 1 -3 4 2 10 -20 11 3 -35 84 -77 29 4 126 -336 396 -261 76 5 -462 1320 -1815 1595 -836 199 6 1716 -5148 7865 -8294 5928 -2587 521 7 -6435 20020 -33033 39585 -34580 20895 -7815 1364 8 24310 -77792 136136 -179452 180880 -135320 70856 -23188 3571 9 -92378 302328 -554268 786828 -883728 771324 -504849 233244 -67849 9349 10 352716 -1175720 2238390 -3372120 4124064 -4049451 3118185 -1814120 749910 -196329 24476 3. R e c u r r e n c e Relations In equation (1.3) of [1], it is noted that (3.1) 3L 2 n + i - L2n-i = L2n+3, n > 2. Using (3.1) with n replaced by n - 1 and (1.1), we see that (3.2) (n + k + l)(n + k)f(n, k) + 3(n + k){n - k + l)/(n, k - 1) + (n - k + l ) ( n - k + 2 ) / ( n , /c - 2) = 0, k > 2.
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