Imagine the sum $latex{S}_n$ as the volume of the pyramid of 1 x 1 x 1 cubes with one cube on the top layer, 4 on the next, 9 on the next and so on up to $latex{n}^2$ cubes on the bottom layer as seen in the figure above. From the above expression, we see that the sum is one sixth the volume of a box with dimensions  $latexn$, $latexn+1$, and $latex2n+1$. So it is at least conceivable that six of these pyramids could be packed into a rectangular volume of that size.  And as the pictures indicate below, there is such an arrangement.

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