Editing Square pyramidal number (section)

==Relations to other figurate numbers==
[[File:Rye Castle, Rye, East Sussex, England-6April2011 (1) (cropped).jpg|thumb|A square pyramid of [[Round shot|cannonballs]] at [[Rye Castle]] in England]]
[[File:Cannonball_problem.svg|thumb|4900 balls arranged as a square pyramid of side 24, and a square of side 70]]
The [[cannonball problem]] asks for the sizes of pyramids of cannonballs that can also be spread out to form a square array, or equivalently, which numbers are both square and square pyramidal.
Besides 1, there is only one other number that has this property: 4900, which is both the 70th square number and the 24th square pyramidal number.{{r|anglin}}

The square pyramidal numbers can be expressed as sums of [[binomial coefficient]]s:{{r|conguy|grassl}}
<math display=block>P_n = \binom{n + 2}{3} + \binom{n + 1}{3} = \binom{n + 1}{2} + 2\binom{n + 1}{3}.</math>

The binomial coefficients occurring in this representation are [[tetrahedral number]]s, and this formula expresses a square pyramidal number as the sum of two tetrahedral numbers in the same way as square numbers split into two consecutive [[triangular number]]s.{{r|beiler|conguy}}

If a tetrahedron is reflected across one of its faces, the two copies form a [[triangular bipyramid]]. The square pyramidal numbers are also the figurate numbers of the triangular bipyramids, and this formula can be interpreted as an equality between the square pyramidal numbers and the triangular bipyramidal numbers.{{r|oeis}} Analogously, reflecting a square pyramid across its base produces an octahedron, from which it follows that each [[octahedral number]] is the sum of two consecutive square pyramidal numbers.{{r|cagbud}}

Square pyramidal numbers are also related to tetrahedral numbers in a different way: the points from four copies of the same square pyramid can be rearranged to form a single tetrahedron with twice as many points along each edge. That is,{{r|alsnel}}
<math display=block>4P_n=Te_{2n}=\binom{2n+2}{3}.</math>

To see this, arrange each square pyramid so that each layer is directly above the previous layer, e.g. the heights are
<pre>
4321
3321
2221
1111
</pre>

Four of these can then be joined by the height {{mvar|4}} pillar to make an even square pyramid, with layers <math>4, 16, 36, \dots</math>.

Each layer is the sum of consecutive triangular numbers, i.e. <math>(1+3), (6+10), (15+21), \dots</math>, which, when totalled, sum to the tetrahedral number.