A particle moves in the Cartesian Plane according to the following rules: 

1.  From any lattice point $ (a,b)$, the particle may only move to $ (a \plus{} 1,b)$, $ (a,b \plus{} 1)$, or $ (a \plus{} 1,b \plus{} 1)$.
2. There are no right angle turns in the particle's path. 

How many different paths can the particle take from $ (0,0)$ to $ (5,5)$?
To solve this problem, we need to count the number of valid paths from \((0,0)\) to \((5,5)\) under the given movement rules and constraints. 

1. **Understanding the Movement Rules:**
   - The particle can move to \((a+1, b)\), \((a, b+1)\), or \((a+1, b+1)\).
   - There are no right-angle turns allowed in the path.

2. **Path Representation:**
   - Each path from \((0,0)\) to \((5,5)\) consists of exactly 5 steps to the right (R), 5 steps up (U), and possibly some diagonal steps (D).
   - The total number of steps is 10, but we need to ensure no right-angle turns.

3. **Counting Valid Paths:**
   - We need to count the number of sequences of R, U, and D such that no right-angle turns occur.
   - A right-angle turn would occur if an R is immediately followed by a U or vice versa.

4. **Dynamic Programming Approach:**
   - Let \(f(a, b)\) be the number of valid paths from \((0,0)\) to \((a,b)\).
   - The recurrence relation considering the movement rules and constraints is:
     \[
     f(a, b) = f(a-1, b) + f(a, b-1) + f(a-1, b-1)
     \]
     with the initial condition \(f(0,0) = 1\).

5. **Implementing the Recurrence:**
   - We can use a 2D array to store the values of \(f(a, b)\) for \(0 \leq a, b \leq 5\).

6. **Calculating the Values:**
   - Initialize \(f(0,0) = 1\).
   - For each point \((a,b)\), calculate \(f(a,b)\) using the recurrence relation.

\[
\begin{array}{c|cccccc}
a \backslash b & 0 & 1 & 2 & 3 & 4 & 5 \\
\hline
0 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & 1 & 2 & 3 & 4 & 5 & 6 \\
2 & 1 & 3 & 6 & 10 & 15 & 21 \\
3 & 1 & 4 & 10 & 20 & 35 & 56 \\
4 & 1 & 5 & 15 & 35 & 70 & 126 \\
5 & 1 & 6 & 21 & 56 & 126 & 252 \\
\end{array}
\]

7. **Final Calculation:**
   - The value at \(f(5,5)\) gives the number of valid paths from \((0,0)\) to \((5,5)\).

\[
f(5,5) = 252
\]

The final answer is \(\boxed{252}\).