<p>You are given an integer <code>finalSum</code>. Split it into a sum of a <strong>maximum</strong> number of <strong>unique</strong> positive even integers.</p>

<ul>
	<li>For example, given <code>finalSum = 12</code>, the following splits are <strong>valid</strong> (unique positive even integers summing up to <code>finalSum</code>): <code>(12)</code>, <code>(2 + 10)</code>, <code>(2 + 4 + 6)</code>, and <code>(4 + 8)</code>. Among them, <code>(2 + 4 + 6)</code> contains the maximum number of integers. Note that <code>finalSum</code> cannot be split into <code>(2 + 2 + 4 + 4)</code> as all the numbers should be unique.</li>
</ul>

<p>Return <em>a list of integers that represent a valid split containing a <strong>maximum</strong> number of integers</em>. If no valid split exists for <code>finalSum</code>, return <em>an <strong>empty</strong> list</em>. You may return the integers in <strong>any</strong> order.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> finalSum = 12
<strong>Output:</strong> [2,4,6]
<strong>Explanation:</strong> The following are valid splits: <code>(12)</code>, <code>(2 + 10)</code>, <code>(2 + 4 + 6)</code>, and <code>(4 + 8)</code>.
(2 + 4 + 6) has the maximum number of integers, which is 3. Thus, we return [2,4,6].
Note that [2,6,4], [6,2,4], etc. are also accepted.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> finalSum = 7
<strong>Output:</strong> []
<strong>Explanation:</strong> There are no valid splits for the given finalSum.
Thus, we return an empty array.
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> finalSum = 28
<strong>Output:</strong> [6,8,2,12]
<strong>Explanation:</strong> The following are valid splits: <code>(2 + 26)</code>, <code>(6 + 8 + 2 + 12)</code>, and <code>(4 + 24)</code>. 
<code>(6 + 8 + 2 + 12)</code> has the maximum number of integers, which is 4. Thus, we return [6,8,2,12].
Note that [10,2,4,12], [6,2,4,16], etc. are also accepted.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= finalSum &lt;= 10<sup>10</sup></code></li>
</ul>

import math

class Solution:
    def maximumEvenSplit(self, finalSum: int) -> List[int]:
        
        # 2,4,6,8,10 -> 40
        # impossible to do in > 6, but we can get to it by 

        # first n evens sum to n(n+1)
        # find largest n such that n(n+1) <= finalSum
        # return 2,4,....2 *(n-1), finalSum - n(n-1)
        # n^2 < n(n + 1) <= finalSum < (n + 1)(n + 2) < (n + 2)^2
        # so floor(math.sqrt(finalSum)) could be n or n + 1
        # n could be floor(math.sqrt(finalSum)) or floor(math.sqrt(finalSum)) - 1

        # finalSum = 4 -> 2, try 2 and 1 2x(2 + 1) = 6 X, 1 x (1 + 1) = 2 - n = 1
        # finalSum = 2 -> 1  try 1 and 0

        if finalSum % 2 == 1:
            return []
        
        n = int(math.sqrt(finalSum))
        if n * (n + 1) > finalSum or (finalSum - n * (n - 1) < 2 * n):
            n -= 1

        out = [2 * x for x in range(1, n)]
        out.append(finalSum - n * (n - 1))
        return out


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