Editing Square pyramidal number (section)

==Geometric enumeration==
<!--
[[File:Squares in a square grid.svg|thumb|upright=0.75|A 5 by 5 square grid, with three of its 55 squares highlighted.]]
-->
[[File:grid_square_count_puzzle.svg|thumb|upright=0.5|All 14 squares in a 3&times;3-square (4&times;4-vertex) grid]]
As well as counting spheres in a pyramid, these numbers can be used to solve several other counting problems. For example, a common [[mathematical puzzle]] involves counting the squares in a large {{mvar|n}} by {{mvar|n}} square grid.{{r|dpas}} This count can be derived as follows:
*The number of {{nowrap|1 &times; 1}} squares in the grid is {{math|''n''<sup>2</sup>}}.
*The number of {{nowrap|2 &times; 2}} squares in the grid is {{math|(''n'' &minus; 1)<sup>2</sup>}}. These can be counted by counting all of the possible upper-left corners of {{nowrap|2 &times; 2}} squares.
*The number of {{math|''k'' &times; ''k''}} squares {{math|(1 ≤ ''k'' ≤ ''n'')}} in the grid is {{math|(''n'' &minus; ''k'' + 1)<sup>2</sup>}}. These can be counted by counting all of the possible upper-left corners of {{math|''k'' &times; ''k''}} squares.

It follows that the number of squares in an {{math|''n'' &times; ''n''}} square grid is:{{r|robitaille}}
<math display=block>n^2 + (n-1)^2 + (n-2)^2 + (n-3)^2 + \ldots = \frac{n(n+1)(2n+1)}{6}.</math>
That is, the solution to the puzzle is given by the {{mvar|n}}-th square pyramidal number.{{r|oeis}} The number of rectangles in a square grid is given by the [[squared triangular number]]s.{{r|stein}}

The square pyramidal number <math>P_n</math> also counts the [[acute triangle]]s formed from the vertices of a <math>(2n+1)</math>-sided [[regular polygon]]. For instance, an equilateral triangle contains only one acute triangle (itself), a regular [[pentagon]] has five acute [[Golden triangle (mathematics)|golden triangle]]s within it, a regular [[heptagon]] has 14 acute triangles of two shapes, etc.{{r|oeis}} More abstractly, when permutations of the rows or columns of a [[Matrix (mathematics)|matrix]] are considered as equivalent, the number of <math>2\times 2</math> matrices with non-negative integer coefficients summing to <math>n</math>, for odd values of <math>n</math>, is a square pyramidal number.{{r|bvt}}