The sum of squares of the first positive integers: . It counts a square-based stack with layers containing objects. Differences give sums over consecutive squares.
The number of objects in a square-based pyramid with successive square layers of sizes . If , direct algebra gives and . Mathematical induction therefore proves the displayed summation formula. Its leading term is , so this sum grows cubically, even though its largest individual summand grows quadratically.
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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 \(1 \times 1\).