Certainly! Below is the Python code that implements the described recursive binary search algorithm:

```python
def binary_search(arr, target, start, end):
    if start > end:
        return -1
    mid = (start + end) // 2
    if arr[mid] == target:
        return mid
    elif arr[mid] < target:
        return binary_search(arr, target, mid + 1, end)
    else:
        return binary_search(arr, target, start, mid - 1)

# Example Usage
arr = [1, 3, 5, 7, 9, 11, 13, 15]
target = 7
result = binary_search(arr, target, 0, len(arr) - 1)
print(result)
```

As for the name of this algorithm, it is called the "Binary Search Algorithm." It is a simple and efficient algorithm for finding a target value within a sorted sequence of elements by repeatedly dividing the search interval in half.