Binet Formulas for Recursive Integer Sequences

Many integer sequences are recursive sequences and can be defined either recursively or explicitly by use of Binet-type formulas. Explorations with Binet’s formula can lead to new and interesting recursive sequences. Among these sequences are the side and diagonal numbers of a square. Many relationships exist between these two sequences in the same way that numerous relationships exist between the Fibonacci and Lucas sequences. This paper provides a technique for generating many Binet formulas and thus creating many formulas that can be proved by mathematical induction to generate their respective recursive sequences. The mathematics in this paper can be described as educational mathematics and as such, occupies a crucial place in undergraduate mathematics education in that the methods employed here foster student gains in logical reasoning, inductive and deductive reasoning, and understanding of mathematical induction. The author has successfully used the mathematical content of this paper in classes in discrete mathematics taken by first-year students in college