Hưởng ứng phong trào trồng cây, công ty X tiến
hành trồng bốn loại cây gồm lim, lộc vừng, bằng lăng và xà cừ trong 7 ngày liên
tiếp. Trong đó, tổ I phụ tách trồng cây lim và lộc vừng, tổ II phụ trách trồng cây
bằng lăng và xà cừ. Hỏi có bao nhiêu cách phân chia các ngày để hai tổ trồng cây
biết rằng mỗi ngày chỉ được trồng một trong bốn loại cây với duy nhất một tổ làm
việc và không có tổ nào làm việc ba ngày liên tiếp ? (Số cây mỗi loại đủ để hai tổ
trồng với mọi cách phân chia các ngày thoả yêu cầu)

To solve this problem, we need to determine the number of ways to assign the days for planting such that each day, only one team (either team I or team II) plants one of the specified trees, with the additional condition that no team works for three consecutive days.

### Step 1: Understand the Problem
- **Teams and Trees**: 
  - Team I plants trees of type Lim and Lộc Vừng.
  - Team II plants trees of type Bằng Lăng and Xà Cừ.
- **Constraints**:
  1. Only one team works each day.
  2. No team can work for three consecutive days.
  3. There are 7 days to be assigned.

### Step 2: Define the problem using a sequence
Let us denote the days when Team I works as `I` and the days when Team II works as `II`. We need to count the number of valid sequences of length 7 made up of `I` and `II` where no three consecutive days are `III` or `III`.

### Step 3: Generate the sequence with restrictions
Let's define:
- \( a_n \): The number of valid sequences of length \( n \).
- \( b_n \): The number of valid sequences of length \( n \) ending with \( I \).
- \( c_n \): The number of valid sequences of length \( n \) ending with \( II \).

Thus, we have the following recurrence relations:
- \( a_n = b_n + c_n \) (total sequences of length \( n \)).
- \( b_n = c_{n-1} + c_{n-2} \) (to avoid three consecutive `I`s).
- \( c_n = b_{n-1} + b_{n-2} \) (to avoid three consecutive `II`s).

### Step 4: Initial conditions
- \( b_1 = 1 \) (sequence `I`).
- \( c_1 = 1 \) (sequence `II`).
- \( b_2 = 1 \) (sequence `II I`).
- \( c_2 = 1 \) (sequence `I II`).

### Step 5: Solve the recurrence for \( n = 7 \)
Compute values of \( b_n \), \( c_n \), and \( a_n \) for \( n = 3 \) to \( n = 7 \):

1. \( b_3 = c_2 + c_1 = 1 + 1 = 2 \)
2. \( c_3 = b_2 + b_1 = 1 + 1 = 2 \)
3. \( a_3 = b_3 + c_3 = 2 + 2 = 4 \)

4. \( b_4 = c_3 + c_2 = 2 + 1 = 3 \)
5. \( c_4 = b_3 + b_2 = 2 + 1 = 3 \)
6. \( a_4 = b_4 + c_4 = 3 + 3 = 6 \)

7. \( b_5 = c_4 + c_3 = 3 + 2 = 5 \)
8. \( c_5 = b_4 + b_3 = 3 + 2 = 5 \)
9. \( a_5 = b_5 + c_5 = 5 + 5 = 10 \)

10. \( b_6 = c_5 + c_4 = 5 + 3 = 8 \)
11. \( c_6 = b_5 + b_4 = 5 + 3 = 8 \)
12. \( a_6 = b_6 + c_6 = 8 + 8 = 16 \)

13. \( b_7 = c_6 + c_5 = 8 + 5 = 13 \)
14. \( c_7 = b_6 + b_5 = 8 + 5 = 13 \)
15. \( a_7 = b_7 + c_7 = 13 + 13 = 26 \)

So, the number of valid ways to assign the days is \( \boxed{26} \).

Would you like further details or have any questions?  
Here are 5 related questions to consider:
1. How would the solution change if we had more than 7 days?
2. What if one of the teams could work for three days in a row?
3. How do we derive the recurrence relations for this type of problem?
4. How would you solve the problem if the constraint of "one type of tree per day" was relaxed?
5. What if both teams could work on the same day, how would that affect the possible sequences?

**Tip:** Understanding and setting up recurrence relations is key to solving combinatorial problems efficiently.

Explore how to count the number of ways to assign planting days for two teams, ensuring no team works for three consecutive days. This combinatorial problem involves sequences and recurrence relations. Suitable for advanced mathematics enthusiasts.

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