16.1.4. Degree of Recurrence Equation

A trivial recurrence condition for my function $s_n$ is the following. I can say that $s_n = s_{n-1} + s_{n-2} + \ldots + s_1$.

This chunk presents a basic recurrence relation for a function denoted as $s_n$. This relation suggests that the current value of the function is the sum of all previous values. Specifically, it states that to find the value of $s_n$, we need to add up all previous terms starting from $s_1$ to $s_{n-1}$. By doing this, we ensure the sequence maintains a pattern that relies on its previous elements.

Think of this relationship like adding scores in a game where each score builds upon the previous ones. If you scored points of 2, 3, and 5 in your first three turns, your total score after those turns is 2 + 3 + 5 = 10. Similarly, each term in our sequence builds on prior terms.

For example, if the second last value in the sequence, namely $a_{n-1}$, can be 1, we find all possible sequences starting with 1 and ending with $n$.

Here, the text discusses sequences that are 'strictly increasing.' When we assume that the second last term is 1, we focus on finding those sequences that begin with 1 and also end with $n$. This demonstrates how specifying certain conditions on previous terms can help in determining the future values in a sequence.

Imagine you are stacking blocks where you can only stack higher numbers on top. If the second last block is numbered 1, the next block must be numbered at least 2. This limitation informs how you can build the stack.

Now, there can be two categories: Category 1 where $a_{n-1}$ is $n - 1$. Category 2 within these sequences allows $a_{n-1}$ to be any number, not equal to $n - 1$, and ending with $n$.

The sequences are categorized based on the second last value $a_{n-1}$. In Category 1, we strictly consider sequences whose second last value is equal to $n - 1$, while Category 2 includes sequences that can have any valid second last value as long as they adhere to the increasing sequence rule preceding n. This categorization helps simplify our understanding and calculations of valid sequences.

Think of sorting books on a shelf by their height. Category 1 represents books that are just one height shorter than the tallest book (our sequence ending with n). In Category 2, you can have books of any height shorter than the tallest, making the possibilities wider.

We want a more compact recurrence condition. So let us derive that alternate recurrence equation...

This section bridges the basic recurrence condition to a more compact one, highlighting the importance of efficiency in defining our recurrence relations. The alternate recurrence equation is derived from analyzing the categories of sequences previously discussed. By observing the sequences' structural forms, we simplify the recurrence equation's complexity. It helps us understand how the current term behaves relative to only a few preceding terms.

Imagine simplifying a recipe that requires all ingredients (previous terms). Instead of listing everything, you could choose a few essential ingredients that still yield a delicious dish. This is analogous to creating a more straightforward mathematical expression that provides the same results with fewer parts.

This is an equation of degree 1 because now the dependency of the n-th term is only on the previous term.

The degree of a recurrence equation indicates how many preceding terms influence the subsequent term. Here, the newly derived equation shows that the n-th term depends solely on the immediately prior term rather than all previous terms. This reduction to degree 1 makes computations simpler because fewer initial conditions are necessary to solve it.

Consider a relay race where only the person directly before you determines how fast you start running. If you only need to focus on one runner instead of the entire team’s performance, you can react more quickly and adjust your strategy accordingly.

It turns out that even though this alternate recurrence equation is of degree 1, we need 2 initial conditions.

Despite the simplicity of a degree 1 equation, it still requires two initial conditions to anchor our sequence correctly. The reasoning here involves recognizing that certain sequences only emerge depending on the values and configuration of the initial conditions provided. As a consequence, each unique starting point impacts the sequence's growth.

Think of planting a garden where the initial two plants you choose significantly affect how the garden grows overall. Each type of plant (or initial condition) sets the stage for what can flourish next, just like our recurrence relations depend on their initial conditions to evolve.

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