Editing Square pyramidal number (section)

==Formula==
[[File:SumSqWM.webm|thumb|Six copies of a square pyramid with {{mvar|n}} steps can fit in a cuboid of size {{math|''n''(''n'' + 1)(2''n'' + 1)}}]]
If spheres are packed into square pyramids whose number of layers is 1, 2, 3, etc., then the square pyramidal numbers giving the numbers of spheres in each pyramid are:{{r|oeis|beiler}}

{{block indent|left=1.6|[[1 (number)|1]], [[5 (number)|5]], [[14 (number)|14]], [[30 (number)|30]], [[55 (number)|55]], [[91 (number)|91]], [[140 (number)|140]], [[204 (number)|204]], [[285 (number)|285]], [[300 (number)#385|385]], 506, 650, 819, ... .}}
These numbers can be calculated algebraically, as follows. If a pyramid of spheres is decomposed into its square layers with a square number of spheres in each, then the total number <math>P_n</math>  of spheres can be counted as the sum of the number of spheres in each square, <math display=block>P_n = \sum_{k=1}^nk^2 = 1 + 4 + 9 + \cdots + n^2,</math>
and this [[summation]] can be solved to give a [[cubic polynomial]], which can be written in several equivalent ways:
<math display=block>P_n= \frac{n(n + 1)(2n + 1)}{6} = \frac{2n^3 + 3n^2 + n}{6} = \frac{n^3}{3} + \frac{n^2}{2} + \frac{n}{6}.</math>
This equation for a sum of squares is a special case of [[Faulhaber's formula]] for sums of powers, and may be proved by [[mathematical induction]].{{r|hmu}}

More generally, [[figurate number]]s count the numbers of geometric points arranged in regular patterns within certain shapes. The centers of the spheres in a pyramid of spheres form one of these patterns, but for many other types of figurate numbers it does not make sense to think of the points as being centers of spheres.{{r|beiler}}
In modern mathematics, related problems of counting points in [[Integral polytope|integer polyhedra]] are formalized by the [[Ehrhart polynomial]]s. These differ from figurate numbers in that, for Ehrhart polynomials, the points are always arranged in an [[integer lattice]] rather than having an arrangement that is more carefully fitted to the shape in question, and the shape they fit into is a polyhedron with lattice points as its vertices. Specifically, the Ehrhart polynomial {{math|''L''(''P'',''t'')}} of an integer polyhedron {{mvar|P}} is a [[polynomial]] that counts the integer points in a copy of {{mvar|P}} that is expanded by multiplying all its coordinates by the number {{mvar|t}}. The usual symmetric form of a square pyramid, with a [[unit square]] as its base, is not an integer polyhedron, because the topmost point of the pyramid, its apex, is not an integer point. Instead, the Ehrhart polynomial can be applied to an asymmetric square pyramid {{mvar|P}} with a unit square base and an apex that can be any integer point one unit above the base plane. For this choice of {{mvar|P}}, the Ehrhart polynomial of a pyramid is {{math|1={{sfrac|(''t'' + 1)(''t'' + 2)(2''t'' + 3)|6}} = ''P''<sub>''t'' + 1</sub>}}.{{r|bddps}}