A square pyramidal number is a figurate number that represents the total number of stacked squares in a pyramid with a square base. The \(n\)-th square pyramidal number counts the number of squares in a pyramid that has \(n\) layers, where the bottom layer is \(n \times n\) and each layer above decreases by 1 in both dimensions until the top layer, which is \(1 \times 1\).
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The th square pyramidal number counts a square-based stack with layers containing objects:For example, . To prove the formula, let . Direct algebra gives and , so mathematical induction identifies with the sum of squares.
Differences give sums over consecutive square numbers. The leading term shows cubic growth of the total, even though the largest layer has only objects.