Let's work through the problems you've uploaded. Here's a breakdown of each:

Problem 2:

We are asked to find a recurrence relation for $a_n$, which counts the number of binary strings of length $n$ where:

• Each block of 0s has even length.
• Each block of 1s has odd length.

For $n = 6$, the possible strings are: $000000, 001001, 100100, 111001, 100111, 100001$

The task is to show that: $a_n = 2a_{n-2} + a_{n-3} - a_{n-4}$

Hint analysis:

The hint suggests that we should observe how strings end (either in 1 or 00) and formulate a system of equations based on that observation. This likely requires breaking the problem into cases based on how the string ends.

Problem 3:

We are given a function $f : \mathbb{N}^2 \to \mathbb{N}$ defined by the recurrence relation: $f(k, 0) = 1 \quad \forall k \geq 0, \quad f(0, \ell) = 1 \quad \forall \ell \geq 0$ and $f(k, \ell) = f(k, \ell-1) + f(k-1, \ell)$

We need to show that: $f(k, \ell) = \binom{k + \ell}{k}$

Hint analysis:

The hint suggests using induction to prove this formula. The recurrence relation resembles Pascal's identity, which governs binomial coefficients, so we can likely prove this by establishing a base case and using induction.

Problem 4:

We define $X_n$ as the set of non-empty subsets of $[n] = \{1, 2, \dots, n\}$ that do not contain consecutive integers. Examples are given for small $n$.

For a finite subset $A \subseteq \mathbb{N}$, we define a function: $f(A) = \prod_{a \in A} a^2$ And we are tasked with proving that $a_n = f(A)$ for some condition (possibly involving the subsets).

Hint analysis:

Induction will likely be needed to generalize the formula for all $n$, as suggested.

Would you like to go step by step through one of these problems, or do you need help with a specific part? Let me know if you'd like detailed solutions or explanations for any of the problems.

Follow-up questions:

1. How do you interpret the binary string rules in Problem 2?
2. Do you need help setting up the induction for Problem 3?
3. Should we focus on understanding the recurrence relations further?
4. What kind of examples would help you with Problem 4?
5. Have you worked with binomial identities before (related to Problem 3)?

Tip:

When working with recurrence relations, break down the problem into smaller cases and explore how the relation holds for small values of $n$, which can help with identifying patterns.