To solve the problem, we need to compute the number of connected subsets of the set $\{1, 2, \ldots, n\}$, denoted as $U_n$. A set $M$ of positive integers is called connected if for any element $x \in M$, at least one of the numbers $x-1$ or $x+1$ is in $M$.

### Part (a): Compute $U_7$

We start by establishing a recurrence relation for $U_n$.

#### Step 1: Establish the recurrence relation
Consider a connected subset $M$ of $\{1, 2, \ldots, n\}$.

Case 1: $n \notin M$
In this case, $M$ is a connected subset of $\{1, 2, \ldots, n-1\}$. There are $U_{n-1}$ such subsets.

Case 2: $n \in M$
For $M$ to be connected, $n-1$ must also be in $M$. Thus, $M$ can be written as $M = N \cup \{n-1, n\}$, where $N$ is a connected subset of $\{1, 2, \ldots, n-2\}$.

Subcase 2.1: $n-2 \notin N$
In this subcase, $N$ is a connected subset of $\{1, 2, \ldots, n-3\}$. There are $U_{n-3}$ such subsets.

Subcase 2.2: $n-2 \in N$
In this subcase, $N$ is a connected subset of $\{1, 2, \ldots, n-2\}$. There are $U_{n-2}$ such subsets.

Combining these cases, we get:
$$U_n = U_{n-1} + U_{n-2} + U_{n-3}$$

#### Step 2: Compute initial values
We need initial values to start the recurrence:
$$U_1 = 1$$
$$U_2 = 2$$
$$U_3 = 4$$

#### Step 3: Compute $U_4$ to $U_7$
Using the recurrence relation:
$$U_4 = U_3 + U_2 + U_1 = 4 + 2 + 1 = 7$$
$$U_5 = U_4 + U_3 + U_2 = 7 + 4 + 2 = 13$$
$$U_6 = U_5 + U_4 + U_3 = 13 + 7 + 4 = 24$$
$$U_7 = U_6 + U_5 + U_4 = 24 + 13 + 7 = 44$$

### Part (b): Find the smallest $n$ such that $U_n \geq 2006$

We continue using the recurrence relation to find the smallest $n$ such that $U_n \geq 2006$.

#### Step 4: Compute $U_8$ to $U_{15}$
$$U_8 = U_7 + U_6 + U_5 = 44 + 24 + 13 = 81$$
$$U_9 = U_8 + U_7 + U_6 = 81 + 44 + 24 = 149$$
$$U_{10} = U_9 + U_8 + U_7 = 149 + 81 + 44 = 274$$
$$U_{11} = U_{10} + U_9 + U_8 = 274 + 149 + 81 = 504$$
$$U_{12} = U_{11} + U_{10} + U_9 = 504 + 274 + 149 = 927$$
$$U_{13} = U_{12} + U_{11} + U_{10} = 927 + 504 + 274 = 1705$$
$$U_{14} = U_{13} + U_{12} + U_{11} = 1705 + 927 + 504 = 3136$$

Since $U_{14} = 3136 \geq 2006$, the smallest $n$ such that $U_n \geq 2006$ is $n = 14$.

The final answer is $n = 14$