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As Yuval Filmus and gnasher729 have pointed out my original equation for the repetition structure is indeed wrong. I believe this to be the true correct answer: $\sum_{i=1}^{n} (\sum_{j=1}^{i^2} j)$ $= \sum_{i=1}^{n} (\frac{i^4+i^2}{2})$ $= \frac{1}{2} \sum_{i=1}^{n}i^4 + \frac{1}{2} \sum_{i=1}^{n}i^2$ $\approx \frac{n^5}{10} + \frac{n^3}{6}$ $\...


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UPDATE: Since posting this question I made some progress on my own. Since this question did not attract any other answers I thought I'd add what I came up with. My conclusion is my original insight into the problem was incomplete. While the "only-advancing-lesser-index-can-increase-area" insight is true, and in the right direction, there is a deeper insight ...


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