Generating Function for a Recurrence Relation

Find the generating function for a sequence given recursively by: $a_n = a_{n-1} + 2a_{n-2} + 3$ with initial terms $a_0 = 2$ and $a_1 = 2$.

In this problem, we are tasked with finding the generating function for a sequence defined by a specific recurrence relation. Generating functions are a powerful tool in discrete mathematics, particularly in the study of sequences and series. They convert problems about sequences into problems about functions, which can often be more easily manipulated with algebraic techniques. In this case, because the sequence is defined recursively, finding a generating function allows us to encapsulate the recursive structure into a single expression.

The problem involves a recurrence relation with constant coefficients, which is a common type of problem within this area. Here, understanding how to derive and use a generating function to solve recurrence relations is critical. The generating function can transform the recursive process into an algebraic one, enabling us to find closed-form solutions or to analyze the behavior of the sequence more easily.

Besides computing the generating function, this exercise reinforces the concept of manipulating series and using initial conditions in critical ways. These skills are foundational in many areas of discrete math, helping students develop a deeper understanding of the nature of sequences, their growth, and their underlying structures.