A sequence of number pairs can be used to generate many interesting examples in teaching mathematics subjects at various levels. It is often used in elementary or middle school mathematics classes to illustrate the concept of "patterns." In this paper the authors present a few interesting ways of using this sequence to form examples for high school or college mathematics courses, such as algebra, geometry, calculus, and linear algebra. Applications of these examples in teaching those courses are discussed, together with suggestions for possible student in-class exercises and discussions. Interesting examples like these may in general help students to enjoy the class more and make the classroom learning process more effective. The applications discussed in this article are applications in teaching: (1) Analytic Geometry; (2) Limits and Mathematical Induction; (3) Calculus; and (4) Linear Algebra. (Contains 1 figure.)