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# ft\_fibonacci

**Objective:**

Create a **recursive function** that returns the **n-th element** of the Fibonacci sequence. The Fibonacci sequence starts with `0, 1, 1, 2, ...`, where each number is the sum of the two preceding ones. The function should return **-1** if the index is negative.

**Turn-in Directory:**

ex04/

**Files to Turn In:**

ft\_fibonacci.c

**Allowed Functions:**

None

**Prototype:**

```c
int ft_fibonacci(int index);
```

***

#### **Detailed Explanation**

The **ft\_fibonacci** function calculates the **n-th element** of the Fibonacci sequence using recursion. The Fibonacci sequence is defined as:

* `F(0) = 0`
* `F(1) = 1`
* `F(n) = F(n-1) + F(n-2)` for `n > 1`

The function returns:

* The **n-th Fibonacci number** if `index` is a non-negative integer.
* **-1** if `index` is negative.

***

#### **Code Implementation**

```c
int ft_fibonacci(int index)
{
    if (index < 0)  // Invalid index (negative number)
        return (-1);
    else if (index == 0)  // Base case: F(0) = 0
        return (0);
    else if (index == 1 || index == 2)  // Base case: F(1) = 1, F(2) = 1
        return (1);
    else  // Recursive case: F(n) = F(n-1) + F(n-2)
        return (ft_fibonacci(index - 1) + ft_fibonacci(index - 2));
}
```

***

#### **Example Usage**

```c
#include <stdio.h>

int ft_fibonacci(int index);

int main(void)
{
    printf("Fibonacci(0): %d\n", ft_fibonacci(0));  // Output: 0
    printf("Fibonacci(1): %d\n", ft_fibonacci(1));  // Output: 1
    printf("Fibonacci(2): %d\n", ft_fibonacci(2));  // Output: 1
    printf("Fibonacci(5): %d\n", ft_fibonacci(5));  // Output: 5
    printf("Fibonacci(10): %d\n", ft_fibonacci(10)); // Output: 55
    printf("Fibonacci(-3): %d\n", ft_fibonacci(-3)); // Output: -1

    return 0;
}
```

***

#### **Edge Cases to Consider**

1. **Negative Index**:
   * Input: `-3` → Returns **-1** (invalid index).
2. **Index 0**:
   * Input: `0` → Returns **0** (`F(0) = 0`).
3. **Index 1 or 2**:
   * Input: `1` or `2` → Returns **1** (`F(1) = 1`, `F(2) = 1`).
4. **Positive Index**:
   * Input: `5` → Returns **5** (`F(5) = 5`).
   * Input: `10` → Returns **55** (`F(10) = 55`).

***

#### **Limitations of ft\_fibonacci**

1. **No Handling of Large Indices**:
   * For large values of `index`, the function may cause a **stack overflow** due to excessive recursion depth.
2. **Inefficiency**:
   * The recursive approach is inefficient for large indices because it recalculates the same Fibonacci numbers multiple times.
3. **No Handling of Overflows**:
   * The function does not handle integer overflows. For large values of `index`, the result may exceed the range of an `int`, leading to undefined behavior.

***

#### **How It Works**

1. **Base Case 1 (Negative Index)**:

   * If `index` is negative, the function immediately returns **-1** (invalid index).

   ```c
   if (index < 0)
       return (-1);
   ```
2. **Base Case 2 (Index 0)**:

   * If `index` is **0**, the function returns **0** (`F(0) = 0`).

   ```c
   else if (index == 0)
       return (0);
   ```
3. **Base Case 3 (Index 1 or 2)**:

   * If `index` is **1** or **2**, the function returns **1** (`F(1) = 1`, `F(2) = 1`).

   ```c
   else if (index == 1 || index == 2)
       return (1);
   ```
4. **Recursive Case**:

   * For any positive integer greater than 2, the function calls itself with `index - 1` and `index - 2` and returns the sum of the two results.

   ```c
   else
       return (ft_fibonacci(index - 1) + ft_fibonacci(index - 2));
   ```

***

#### **Key Points to Remember**

1. **Recursive Approach**:
   * The function uses recursion to calculate the Fibonacci number, calling itself with smaller indices until it reaches the base cases.
2. **Base Cases**:
   * The base cases handle edge cases like `index = 0`, `index = 1`, and `index = 2`.
3. **Efficiency**:
   * The time complexity is **O(2^n)**, where `n` is the index. This is because the function recalculates the same Fibonacci numbers multiple times.


---

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