`f_(n)={{:(f_(n-1)" if n is even"),(2f_(n-1)" if n is odd"):}`
and `f_(0)=1`, find the value of `f_(16)`.

Answer

To solve the problem, we need to evaluate the function \( f(n) \) defined by the following rules:

- If \( n \) is even, then \( f(n) = f(n-1) \)
- If \( n \) is odd, then \( f(n) = 2 \cdot f(n-1) \)
- The base case is \( f(0) = 1 \)

We are tasked with finding \( f(16) \).

### Step-by-Step Solution:

1. **Calculate \( f(0) \)**:
\[ f(0) = 1 \]

2. **Calculate \( f(1) \)** (since 1 is odd):
\[ f(1) = 2 \cdot f(0) = 2 \cdot 1 = 2 \]

3. **Calculate \( f(2) \)** (since 2 is even):
\[ f(2) = f(1) = 2 \]

4. **Calculate \( f(3) \)** (since 3 is odd):
\[ f(3) = 2 \cdot f(2) = 2 \cdot 2 = 4 \]

5. **Calculate \( f(4) \)** (since 4 is even):
\[ f(4) = f(3) = 4 \]

6. **Calculate \( f(5) \)** (since 5 is odd):
\[ f(5) = 2 \cdot f(4) = 2 \cdot 4 = 8 \]

7. **Calculate \( f(6) \)** (since 6 is even):
\[ f(6) = f(5) = 8 \]

8. **Calculate \( f(7) \)** (since 7 is odd):
\[ f(7) = 2 \cdot f(6) = 2 \cdot 8 = 16 \]

9. **Calculate \( f(8) \)** (since 8 is even):
\[ f(8) = f(7) = 16 \]

10. **Calculate \( f(9) \)** (since 9 is odd):
\[ f(9) = 2 \cdot f(8) = 2 \cdot 16 = 32 \]

11. **Calculate \( f(10) \)** (since 10 is even):
\[ f(10) = f(9) = 32 \]

12. **Calculate \( f(11) \)** (since 11 is odd):
\[ f(11) = 2 \cdot f(10) = 2 \cdot 32 = 64 \]

13. **Calculate \( f(12) \)** (since 12 is even):
\[ f(12) = f(11) = 64 \]

14. **Calculate \( f(13) \)** (since 13 is odd):
\[ f(13) = 2 \cdot f(12) = 2 \cdot 64 = 128 \]

15. **Calculate \( f(14) \)** (since 14 is even):
\[ f(14) = f(13) = 128 \]

16. **Calculate \( f(15) \)** (since 15 is odd):
\[ f(15) = 2 \cdot f(14) = 2 \cdot 128 = 256 \]

17. **Calculate \( f(16) \)** (since 16 is even):
\[ f(16) = f(15) = 256 \]

Thus, the value of \( f(16) \) is \( \boxed{256} \).