ON WEIGHTED r-GENERALIZED FIBONACCI SEQUENCES Francois Dubeau Departement de Mathematiques et dThformatique, Faculte des Sciences, Universite de Sherbrooke, Sherbrooke (Quebec), Canada, J1K 2R1 francois.dubeau@dmi.usherb.ca Walter Motta Departamento de Matematica, CETEC-UFU, Campus Santa Monica, 38400-902 Uberlandia, MG, Brazil wmotta@ufu.br Mustapha Rachidi Departement de Mathematiques, Faculte des Sciences, Universite Mohammed V, B.P. 1014, Rabat, Morocco rachidi@emi.ma Osamu Saeki Department of Mathematics, Faculty of Science, Hiroshima University, Higasm-Hiroshima 739, Japan saeki@top2.mam.sci.hiroshima-u.ac.jp (Submitted August 1994) 1. INTRODUCTION The standard Fibonacci numbers have several well-known and familiar properties, among which are the fact that the ratio of successive terms approaches a fixed limit (/>, and that the n th Fibonacci number is asymptotic to (/)". In this paper we extend these properties to a generalized class of Fibonacci sequences, giving necessary and sufficient conditions for such a sequence to be asymptotic to one of the form n v ~ l X l . In that case, we show how to compute the limiting ratio between the solution and n v ~ l A n , as well as proving that the ratio of successive terms of a solution must have X as a limit. The necessary and sufficient conditions mentioned above are stated in terms of the roots of a polynomial. Indeed, this polynomial is the characteristic polynomial associated with the difference equation defining a generalized Fibonacci sequence. We also discuss conditions that depend directly on the coefficients of the characteristic polynomial. As a special case, we derive results when the polynomial has negative real coefficients (except for the leading coefficient 1). More generally, we give a sufficient condition on the coefficients for the roots to satisfy the necessary and sufficient conditions discussed above. 2. PRELIMINARIES Let a x ,a 2 ,...,a r be arbitrary r > 2 complex numbers with a r & 0, and let A = (a_ r+l , a_ r+2 , ..., <2_i, a 0 ) be any given sequence of complex numbers. The weighted r-generalizedFibonacci sequence {y^(w)}^_ r+1 is the sequence generated by the difference equation with initial values: 102 [MAY