Square pyramidal number
In mathematics, a pyramid number, or square pyramidal number, is a figurate number that represents the number of stacked spheres in a pyramid with a square base. The square pyramidal numbers can be used to count number of squares in an n × n grid, or acute triangles in an odd regular polygon. They equal the sums of consecutive tetrahedral numbers, and are one-fourth of a larger tetrahedral number. The sum of two consecutive square pyramidal numbers is an octahedral number.
The first few square pyramidal numbers are:
These numbers can be expressed in a formula as
In modern mathematics, figurate numbers are formalized by the Ehrhart polynomials. The Ehrhart polynomial L(P,t) of a polyhedron P is a polynomial that counts the number of integer points in a copy of P that is expanded by multiplying all its coordinates by the number t. The Ehrhart polynomial of a pyramid whose base is a unit square with integer coordinates, and whose apex is an integer point at height one above the base plane, is (t + 1)(t + 2)(2t + 3)/ = Pt + 1.
- The number of 1 × 1 boxes found in the grid is n2.
- The number of 2 × 2 boxes found in the grid is (n − 1)2. These can be counted by counting all of the possible upper-left corners of 2 × 2 boxes.
- The number of k × k boxes (1 ≤ k ≤ n) found in the grid is (n − k + 1)2. These can be counted by counting all of the possible upper-left corners of k × k boxes.
It follows that the number of squares in an n × n square grid is:
The square pyramidal number also counts the number of acute triangles formed from the vertices of a -sided regular polygon. For instance, an equilateral triangle contains only one acute triangle (itself), a regular pentagon has five acute golden triangles within it, a regular heptagon has 14 acute triangles of two shapes, etc.
Relations to other figurate numbers
The cannonball problem asks 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. This fact was proven by G. N. Watson in 1918.
The square pyramidal numbers can be expressed as sums of binomial coefficients:
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 of slightly more than twice the edge length. That is,
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