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Huh. I just noticed that the sums for n = 16 have a funny pattern... The solution for this is {16, 9, 7, 2, 14, 11, 5, 4, 12, 13, 3, 6, 10, 15, 1, 8}; however, if we take the sequence of "in-between" square sums, we have: {25, 16, 9, 16, 25,...