Recurrence Condition

Today, we're going to dive into recurrence relations, particularly how they relate to sequences. Can anyone tell me what a recurrence relation is?

Isn’t it a way to define a sequence where each term is based on previous ones?

Exactly! A recurrence relation defines terms based on earlier terms. For our case, we're examining sequences that are strictly increasing. Can anyone think of an example of a strictly increasing sequence?

Like 1, 2, 3, 4?

Yes! That’s a perfect example. For our recurrence relation, we denote this number of sequences as $S(n)$. But how might you derive how many of these sequences exist?

We could look at what sequences end with a specific number?

Absolutely! We can break it down based on the last term. If the last term is n, which sequences could precede it? We categorize these based on the value before the last term.

So we would consider the second last value?

Correct! This gives us our first recurrence relation: $S(n) = S(n-1) + S(n-2) + ... + S(1)$. Remember, this can get cumbersome with lots of prior values. Let’s summarize what we just discussed.

Now that we've established the basic recurrence, let’s make it more compact. Instead of relying on many previous terms, let’s focus on just the last one—how do we do that?

Maybe we could reinterpret the references so they all link back to the last term?

That’s a great idea! If we consider our sequences that end with 'n', we can break them down into two categories: when the second last value is n - 1 and when it isn’t. This leads us to: $S(n) = 2 * S(n-1)$.

So, if I understand correctly, we only have to worry about the last value now?

That's right! And with this new formulation, we simplify our calculations significantly. What do you think we still need to ensure our recurrence works accurately?

We need initial conditions!

Exactly! We need to set firm initial conditions like $S(1) = 1$ and $S(2) = 1$ to anchor our recurrence. Let’s summarize what we learned today.

Now that we have the recurrence relations, let’s put this into practice. Suppose we want to count the bitstrings of length n that contain three zeros in a row. How would we apply our recurrence method here?

We could create a new recurrence for the count of bitstrings without ‘000’ instead!

Great observation! By focusing on the sequences that do NOT contain ‘000’, we can derive and manipulate recurrences effectively to find our answer. Can someone express how these sequences would work?

So we'd define sequences based on the absence of bad patterns?

Precisely! Once we define such sequences, we can sum them over categories just like we did with strictly increasing sequences. Who can summarize our discussion?

We find the recurrence relation from sequences leading into the count, focusing on how many sequences fit certain conditions beginning with valid patterns!

Awesome summary! Understanding how to apply these principles opens up countless applications in analyzing sequences.