Volume 80 BULLETIN of the May 2017 INSTITUTE of COMBINATORICS and its APPLICATIONS Editors-in-Chief: Marco Buratti, Don Kreher, Tran van Trung Boca Raton, Florida ISSN: 1183-1278 BULLETIN OF THE ICA Volume 80 (2017), Pages 99{118 Horadam Sequences: A Survey Update and Extension Peter J. Larcombe Department of Computing and Mathematics College of Engineering and Technology University of Derby Kedleston Road, Derby DE22 1GB, U.K.
[email protected] Abstract We give an update on work relating to Horadam sequences that are generated by a general linear recurrence formula of order two. This article extends a first ever survey published in early 2013 in this Bulletin, and includes coverage of a new research area opened up in recent times. 1 Introduction 1.1 Preamble Looking back over time we can see that, within the vast and ill-defined field of applied mathematics, the advent of serious computing power in the 1970s began to accelerate a then already discernible breach between classic analytical mathematics (concentrating on problems and theories involving continuous phenomena) and that which tackled discrete concepts and en- tities. The latter has, over the last sixty years or so, developed its own characteristics and subject personality—aided en route by increased expo- sure to software tools/languages—and it is in this broad sphere of what 99 Accepted: 15 March 2017 is nowadays known as discrete mathematics that the theory of integer se- quences lies, with many applications now found in number theory and com- binatorics to name but two