Algebraic and Geometric Aspects of some Binary Recurrence Sequences

Abstract

The class of binary recurrence relations is the mother of many important integer sequences. Fibonacci and Lucas sequences are solutions of a single binary recurrence, though the initial values are different. A class of binary recurrence relations generates balancing-like and Lucas-balancing-like sequences. In addition, it also generates the balancing and Lucas-balancing sequences and the even indexed terms of the Fibonacci sequence. The sequence of natural numbers is also generated by a member of this class. The balancing numbers have several generalizations such as the cobalancing numbers, sequence balancing and cobalancing numbers, gap balancing numbers, almost balancing and cobalancing numbers etc. The almost balancing and cobalancing numbers admit further generalizations to super- and sub- balancing and cobalancing numbers. An important observation about the balancing sequence is that the second balancing number is perfect. Unfortunately, no other balancing number is perfect, nor does the cobalancing sequence contain any perfect number. There are interesting sum formulas associated with the balancing and Lucas-balancing sequence; some of these formulas resemble sum formulas for the natural numbers. The exact values for certain infinite sums involving reciprocals of balancing and Lucas-balancing numbers are very difficult to calculate; however, sharp bounds for these sums are expressible in terms of balancing and Lucas-balancing numbers. There are many geometric aspects of sequences associated with binary recurrences. Certain pairs of triangles in the plane with coordinates of vertices as Fibonacci numbers have the important properties such as orthology, paralogy and reverse similarity. Many important identities and assertions about balancing numbers can be proved visually by the help of discrete geometry.