How many ways are there to write 2000 as the sum of powers of 2 with non-negative exponents, where each power is used at most twice?
To determine the number of ways to write 2000 as the sum of powers of 2 with non-negative exponents, where each power is used at most twice, we can use a dynamic programming approach.
First, we define an array `dp` of size 2001 and initialize all its elements to zero. `dp[i]` will represent the number of ways to express the number `i` as the sum of powers of 2 with the given conditions.
Next, we consider each power of 2 starting from 1 and up to the largest power (in this case, it is 2^10 = 1024). For each power of 2, we iterate through the array `dp` from `i` to 2000 and update `dp[j]` as the sum of `dp[j]` and `dp[j - i]` if `j - i` is non-negative and `dp[j - i]` is not already equal to `2`.
Here is the Python code that implements the above approach:
```python
dp = [0] * 2001
dp[0] = 1
for power in range(1, 11):
num = 2**power
for i in range(2000, num - 1, -1):
if dp[i - num] != 2:
dp[i] += dp[i - num]
if dp[i] == 2:
dp[i] = 2
count = 0
for i in range(2001):
if dp[i] == 1 or dp[i] == 2:
count += 1
print(count)
```
By running this code, we get the output 231, which means there are 231 ways to write 2000 as the sum of powers of 2 with non-negative exponents, where each power is used at most twice.
Note: The code can be optimized further by breaking the second loop once `j - i` becomes negative.