Let <code>dp[x]</code> represent the maximum sum of a balanced subsequence ending at <code>x</code>.

Rewriting the formula <code>nums[i_j] - nums[i_j-1] >= i_j - i_j-1</code> gives <code>nums[i_j] - i_j >= nums[i_j-1] - i_j-1</code>.

So, for some index <code>x</code>, we need to find an index <code>y</code>, <code>y < x</code>, such that <code>dp[x] = nums[x] + dp[y]</code> is maximized, and <code>nums[x] - x >= nums[y] - y</code>.

There are many ways to achieve this. One method involves sorting the values of <code>nums[x] - x</code> for all indices <code>x</code> and using a segment/Fenwick tree with coordinate compression.

Hence, using a dictionary or map, let's call it <code>dict</code>, where <code>dict[nums[x] - x]</code> represents the position of the value, <code>nums[x] - x</code>, in the segment tree.

The tree is initialized with zeros initially.

For indices <code>x</code> in order from <code>[0, n - 1]</code>, <code>dp[x] = max(nums[x]</code>, <code>nums[x]</code> + the maximum query from the tree in the range <code>[0, dict[nums[x] - x]])</code>, and if <code>dp[x]</code> is greater than the value in the tree at position <code>dict[nums[x] - x]</code>, we update the value in the tree.

The answer to the problem is the maximum value in <code>dp</code>.