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Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: EXTENDING GENERALIZED FIBONACCI SEQUENCES AND THEIR BINET-TYPE FORMULA | EXTENDING GENERALIZED FIBONACCI SEQUENCES AND THEIR BINET-TYPE FORMULA MUSTAPHA RACHIDI AND OSAMU SAEKI Received 10 March 2006 Accepted 2 July 2006 We study the extension problem of a given sequence defined by a finite order recurrence to a sequence defined by an infinite order recurrence with periodic coefficient sequence. We also study infinite order recurrence relations in a strong sense and give a complete answer to the extension problem. We also obtain a Binet-type formula answering several open questions about these sequences and their characteristic power series. Copyright 2006 M. Rachidi and O. Saeki. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction The notion of an 00 -generalized Fibonacci sequence to-GFS has been introduced in 7 and studied in 1 8 10 . This class of sequences defined by linear recurrences of infinite order is an extension of the class of ordinary weighted r-generalized Fibonacci sequences r-GFSs with r finite defined by linear recurrences of rth order e.g. see 3-6 9 etc. . Such sequences are defined as follows. Let ai 0 0 and a-i 0 0 be two sequences of complex numbers where ai 0 for some i. The associated 00-GFS Vn n Ei is defined by Vn an if n 0 1.1 Vn Ỉ aiVn-i-1 if n 1. 1.2 i 0 The sequences ai 0 0 and a-i 0 0 are called the coefficient sequence and the initial sequence respectively. As is easily observed the general terms Vn may not necessarily exist. In 1 necessary and sufficient conditions for the existence of the general terms have been studied. When there exists an r 1 such that ai 0 for all i r we call the sequence Vn n -r 1 an r-GFS with initial sequence V-r 1 V-r 2 . V0 . For an r-GFS Hindawi Publishing Corporation Advances in Difference Equations Volume 2006 Article ID 23849 Pages 1-11 DOI 10.1155 ADE 2006 23849 2 Extending generalized Fibonacci .
