A frog is travelling from point A (0,0) to point B (4,6) but each step can only be 1 unit up or 1 unit to the right. Additionally, the frog refuses to move three steps in the same direction consecutively. Compute the number of ways the frog can move from A to B.

To solve the problem of computing the number of ways the frog can move from point A (0,0) to point B (4,6) with the given constraints, we need to consider both the paths and the restriction that the frog cannot take three consecutive steps in the same direction. Let's denote the steps to the right as \(R\) and the steps up as \(U\).

Given:
- The frog needs to take 4 steps to the right (R) and 6 steps up (U).
- No three consecutive steps can be in the same direction.

### Step 1: Calculate the Total Number of Unrestricted Paths
Without any restrictions, the total number of paths is the number of ways to arrange 4 \(R\)s and 6 \(U\)s:
\[
\binom{10}{4} = \frac{10!}{4! \cdot 6!} = 210
\]

### Step 2: Account for the Constraint
To incorporate the restriction, we need to use dynamic programming to count valid sequences where no three consecutive steps are in the same direction.

Define \(dp(i, j, k)\) as the number of valid sequences to reach the point \((i, j)\) where the last step was \(k\) (either \(k=0\) for right or \(k=1\) for up), and we keep track of whether the last two steps were in the same direction.

#### Initialization
We start at \((0, 0)\):
\[
dp(0, 0, 0) = dp(0, 0, 1) = 1
\]

#### Recurrence Relations
For moving right from \((i, j)\):
\[
dp(i, j, 0) = dp(i-1, j, 1) + \left\{
\begin{array}{ll}
dp(i-2, j, 1) & \text{if } i > 1 \\
0 & \text{if } i \leq 1
\end{array}
\right.
\]

For moving up from \((i, j)\):
\[
dp(i, j, 1) = dp(i, j-1, 0) + \left\{
\begin{array}{ll}
dp(i, j-2, 0) & \text{if } j > 1 \\
0 & \text{if } j \leq 1
\end{array}
\right.
\]

#### Final Computation
We need to calculate \(dp(4, 6, 0)\) and \(dp(4, 6, 1)\) by iterating through all valid states using the above recurrence relations.

### Dynamic Programming Table
We'll build the dynamic programming table from \((0, 0)\) to \((4, 6)\):

```python
# Initialize dp table
dp = [[[[0, 0] for _ in range(7)] for _ in range(5)] for _ in range(2)]

# Starting point
dp[0][0][0][0] = dp[0][0][0][1] = 1

# Fill the table
for i in range(5):
    for j in range(7):
        if i > 0:
            dp[1][i][j][0] = dp[0][i-1][j][1] + (dp[0][i-2][j][1] if i > 1 else 0)
        if j > 0:
            dp[1][i][j][1] = dp[0][i][j-1][0] + (dp[0][i][j-2][0] if j > 1 else 0)
        # Carry forward previous states
        dp[0][i][j][0] = dp[1][i][j][0]
        dp[0][i][j][1] = dp[1][i][j][1]

# Result
total_ways = dp[1][4][6][0] + dp[1][4][6][1]
print(total_ways)
```

### Result
The number of valid ways the frog can move from \((0,0)\) to \((4,6)\) while adhering to the restriction is the sum of the valid paths ending with a right step and the valid paths ending with

Explore how to compute the number of ways a frog can move from point (0,0) to (4,6) under constraints. This problem involves combinatorial paths and dynamic programming, suitable for advanced high school students.

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