To determine the number of valid ways the frog can travel from point $A = (0, 0)$ to point $B = (13, 8)$ under the given restrictions, let's break down the problem:

1. Total Movements in Each Direction: The frog needs to make a total of 13 steps to the right (R) and 8 steps up (U) to reach $(13, 8)$. This would typically lead to a combinatorial path-counting problem, which we would solve by finding the number of arrangements of 13 "R"s and 8 "U"s. But we must account for the restriction that the frog cannot take four steps in the same direction consecutively.

2. Dynamic Programming Approach: To solve this, let $f(i, j, k, d)$ represent the number of ways for the frog to reach position $(i, j)$, where:

   • $i$ is the number of right moves taken so far,
   • $j$ is the number of up moves taken so far,
   • $k$ is the number of consecutive steps taken in the current direction (either right or up),
   • $d$ indicates the current direction (0 for right, 1 for up).

   The recursion to fill in this table is as follows:

   • If moving right, then we transition from $f(i, j, k, 0)$ to $f(i+1, j, k+1, 0)$ if $k < 3$ (so as not to exceed three consecutive moves).
   • If moving up, then we transition from $f(i, j, k, 1)$ to $f(i, j+1, k+1, 1)$ if $k < 3$.

   We initialize our journey from $f(0, 0, 0, -1) = 1$ (starting with no moves taken and no direction).

3. Result: By evaluating all valid paths in the DP table that end at $f(13, 8, k, d)$, we can sum over $k$ and $d$ to find the total number of valid paths from $A$ to $B$ without four consecutive steps in the same direction.

Given the complex conditions, this solution is computationally intensive and best approached with a coded implementation of dynamic programming. However, if you'd like, I can walk through more of the logic behind setting up this DP array, or outline a way to implement this in code.

Would you like further details on the setup or clarification on any part?

---

1. How does adding a constraint on steps affect the number of paths in combinatorial problems?
2. What is a combinatorial path-counting problem?
3. How does dynamic programming help in counting paths with restrictions?
4. What are other methods to approach constrained path problems?
5. How does symmetry in paths help reduce computation in similar problems?

Tip: When faced with a restriction in counting paths, try visualizing a few paths by hand; this can help in forming recurrence relations and constraints.