In this book we consider the concept of sequences in mathematics. We give a brief discussion of some know concepts and theorems like "Sarkoviskii's Theorem". Recently, many authors consider new concepts like rational sequences and piece-wise defined sequences. We talk about this new trend and develop our own concepts. We prove many results regarding these new concepts. Specifically, we focus on the sequences defined as Q_{k}=hQ_{k-1}² , if Q_{k-1}≤v, Q_{k}=hQ_{k-1}-j , if Q_{k-1}>v, for k=1,2,... . where h, j, v, and Q₀ are fixed real numbers. We and J. Guyker pubilshed a paper entitled "Nonlinear Piecewise De…fined Difference Equations" in International Mathematical Forum, Vol. 7, 2012, no. 1, 47 - 56.