What is the Maximum Sum of Subsequence with No Forty-Eight Consecutive Elements problem and how can it be solved using Dynamic Programming?

What is the Maximum Sum of Subsequence with No Forty-Eight Consecutive Elements problem and how can it be solved using Dynamic Programming?

The Maximum Sum of Subsequence with No Forty-Eight Consecutive Elements problem is a dynamic programming problem that involves finding the maximum sum of a subsequence from a given sequence, with the constraint that no two consecutive elements in the subsequence can be equal to 48.

To solve this problem using dynamic programming, we can use an array to store the maximum sum of subsequences ending at each index. Let's call this array dp.

We initialize dp[0] as the first element of the sequence, and dp[1] as the maximum value between the first and second elements of the sequence. For all other indices i greater than 1, we calculate dp[i] as the maximum value between the sum of the current element and dp[i-2], and dp[i-1].

This is because if we include the current element in the subsequence, we cannot include the previous element, so the maximum sum would be the sum of the current element and the maximum sum of the subsequence ending at i-2. On the other hand, if we exclude the current element, the maximum sum would be the same as the maximum sum of the subsequence ending at i-1.

Finally, the maximum sum of the subsequence with no forty-eight consecutive elements would be the maximum value in the dp array.

By using this dynamic programming approach, we can efficiently solve the Maximum Sum of Subsequence with No Forty-Eight Consecutive Elements problem.