Square pyramidal number

Last updated Geometric representation of the square pyramidal number 1 + 4 + 9 + 16 = 30.

In mathematics, a pyramid number, or square pyramidal number, is a natural number that counts the number of stacked spheres in a pyramid with a square base. The study of these numbers goes back to Archimedes and Fibonacci. They are part of a broader topic of figurate numbers representing the numbers of points forming regular patterns within different shapes.

Contents

As well as counting spheres in a pyramid, these numbers can be described algebraically as a sum of the first $n$ positive square numbers, or as the values of a cubic polynomial. They can be used to solve several other counting problems, including counting squares in a square grid and counting acute triangles formed from the vertices of 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.

History

The pyramidal numbers were one of the few types of three-dimensional figurate numbers studied in Greek mathematics, in works by Nicomachus, Theon of Smyrna, and Iamblichus.  Formulas for summing consecutive squares to give a cubic polynomial, whose values are the square pyramidal numbers, are given by Archimedes, who used this sum as a lemma as part of a study of the volume of a cone,  and by Fibonacci, as part of a more general solution to the problem of finding formulas for sums of progressions of squares.  The square pyramidal numbers were also one of the families of figurate numbers studied by Japanese mathematicians of the wasan period, who named them "kirei saijo suida". 

The same problem, formulated as one of counting the cannonballs in a square pyramid, was posed by Walter Raleigh to mathematician Thomas Harriot in the late 1500s, while both were on a sea voyage. The cannonball problem, asking whether there are any square pyramidal numbers that are also square numbers other than 1 and 4900, is said to have developed out of this exchange. Édouard Lucas found the 4900-ball pyramid with a square number of balls, and in making the cannonball problem more widely known, suggested that it was the only nontrivial solution.  After incomplete proofs by Lucas and Claude-Séraphin Moret-Blanc, the first complete proof that no other such numbers exist was given by G. N. Watson in 1918. 

Formula

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:  

1, 5, 14, 30, 55, 91, 140, 204, 285, 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 $P_{n}$ of spheres can be counted as the sum of the number of spheres in each square,

$P_{n}=\sum _{k=1}^{n}k^{2}=1+4+9+\cdots +n^{2},$ and this summation can be solved to give a cubic polynomial, which can be written in several equivalent ways:

$P_{n}={\frac {n(n+1)(2n+1)}{6}}={\frac {2n^{3}+3n^{2}+n}{6}}={\frac {n^{3}}{3}}+{\frac {n^{2}}{2}}+{\frac {n}{6}}.$ 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. 

More generally, figurate numbers 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.  In modern mathematics, related problems of counting points in integer polyhedra are formalized by the Ehrhart polynomials. 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 L(P,t) of an integer 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 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 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 P, the Ehrhart polynomial of a pyramid is (t + 1)(t + 2)(2t + 3)/6 = Pt + 1. 

Geometric enumeration

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 finding the number of squares in a large n by n square grid.  This number can be derived as follows:

• The number of 1 × 1 squares found in the grid is n2.
• The number of 2 × 2 squares found in the grid is (n 1)2. These can be counted by counting all of the possible upper-left corners of 2 × 2 squares.
• The number of k×k squares (1 ≤ kn) found in the grid is (nk + 1)2. These can be counted by counting all of the possible upper-left corners of k×k squares.

It follows that the number of squares in an n×n square grid is: 

$n^{2}+(n-1)^{2}+(n-2)^{2}+(n-3)^{2}+\ldots ={\frac {n(n+1)(2n+1)}{6}}.$ That is, the solution to the puzzle is given by the nth square pyramidal number.  The number of rectangles in a square grid is given by the squared triangular numbers. 

The square pyramidal number $P_{n}$ also counts the number of acute triangles formed from the vertices of a $(2n+1)$ -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.  More abstractly, when permutations of the rows or columns of a matrix are considered as equivalent, the number of $2\times 2$ matrices with non-negative integer coefficients summing to $n$ , for odd values of $n$ , is a square pyramidal number. 

Relations to other figurate numbers

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. 

The square pyramidal numbers can be expressed as sums of binomial coefficients:  

$P_{n}={\binom {n+2}{3}}+{\binom {n+1}{3}}.$ The binomial coefficients occurring in this representation are tetrahedral numbers, and this formula expresses a square pyramidal number as the sum of two tetrahedral numbers in the same way as square numbers are the sums of two consecutive triangular numbers.   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.  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. 

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, 

$4P_{n}=T_{2n}={\binom {2n+2}{3}}.$ Other properties

The alternating series of unit fractions with the square pyramidal numbers as denominators is closely related to the Leibniz formula for π, although it converges more quickly. It is: 

{\begin{aligned}\sum _{i=1}^{\infty }&(-1)^{i-1}{\frac {1}{P_{i}}}\\&=1-{\frac {1}{5}}+{\frac {1}{14}}-{\frac {1}{30}}+{\frac {1}{55}}-{\frac {1}{91}}+{\frac {1}{140}}-{\frac {1}{204}}+\cdots \\&=6(\pi -3)\\&\approx 0.849556.\\\end{aligned}} In approximation theory, the sequences of odd numbers, sums of odd numbers (square numbers), sums of square numbers (square pyramidal numbers), etc., form the coefficients in a method for converting Chebyshev approximations into polynomials. 

Related Research Articles

In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no perfect square other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, 10 = 2 ⋅ 5 is square-free, but 18 = 2 ⋅ 3 ⋅ 3 is not, because 18 is divisible by 9 = 32. The smallest positive square-free numbers are

In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in India, Persia, China, Germany, and Italy. A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The nth triangular number is the number of dots in the triangular arrangement with n dots on each side, and is equal to the sum of the n natural numbers from 1 to n. The sequence of triangular numbers, starting with the 0th triangular number, is In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals 32 and can be written as 3 × 3.

In mathematics, a polygonal number is a number represented as dots or pebbles arranged in the shape of a regular polygon. The dots are thought of as alphas (units). These are one type of 2-dimensional figurate numbers.

In mathematics, an integral polytope has an associated Ehrhart polynomial that encodes the relationship between the volume of a polytope and the number of integer points the polytope contains. The theory of Ehrhart polynomials can be seen as a higher-dimensional generalization of Pick's theorem in the Euclidean plane.

The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes and different dimensions. The term can mean In geometry, Pick's theorem provides a formula for the area of a simple polygon with integer vertex coordinates, in terms of the number of integer points within it and on its boundary. The result was first described by Georg Alexander Pick in 1899. It was popularized in English by Hugo Steinhaus in the 1950 edition of his book Mathematical Snapshots. It has multiple proofs, and can be generalized to formulas for certain kinds of non-simple polygons. A pyramidal number is a figurate number that represents a pyramid with a polygonal base and a given number of triangular sides. A pyramidal number is the number of points in a pyramid where each layer of the pyramid is an r-sided polygon of points. The term often refers to square pyramidal numbers, which have a square base with four sides, but it can also refer to pyramids with three or more sides. The numbers of points in the base are given by polygonal numbers of the given number of sides, while the numbers of points in each triangular side is given by a triangular number. It is possible to extend the pyramidal numbers to higher dimensions. A tetrahedral number, or triangular pyramidal number, is a figurate number that represents a pyramid with a triangular base and three sides, called a tetrahedron. The nth tetrahedral number, Ten, is the sum of the first n triangular numbers, that is,

An octagonal number is a figurate number that represents an octagon. The octagonal number for n is given by the formula 3n2 - 2n, with n > 0. The first few octagonal numbers are: In number theory, an octahedral number is a figurate number that represents the number of spheres in an octahedron formed from close-packed spheres. The nth octahedral number can be obtained by the formula: A pronic number is a number which is the product of two consecutive integers, that is, a number of the form n(n + 1). The study of these numbers dates back to Aristotle. They are also called oblong numbers, heteromecic numbers, or rectangular numbers; however, the term "rectangular number" has also been applied to the composite numbers. In elementary number theory, a centered square number is a centered figurate number that gives the number of dots in a square with a dot in the center and all other dots surrounding the center dot in successive square layers. That is, each centered square number equals the number of dots within a given city block distance of the center dot on a regular square lattice. While centered square numbers, like figurate numbers in general, have few if any direct practical applications, they are sometimes studied in recreational mathematics for their elegant geometric and arithmetic properties. In geometry, close-packing of equal spheres is a dense arrangement of congruent spheres in an infinite, regular arrangement. Carl Friedrich Gauss proved that the highest average density – that is, the greatest fraction of space occupied by spheres – that can be achieved by a lattice packing is A pentatope number is a number in the fifth cell of any row of Pascal's triangle starting with the 5-term row 1 4 6 4 1, either from left to right or from right to left. In number theory, the sum of the first n cubes is the square of the nth triangular number. That is, In the mathematics of figurate numbers, the cannonball problem asks which numbers are both square and square pyramidal. The problem can be stated as: given a square arrangement of cannonballs, for what size squares can these cannonballs also be arranged into a square pyramid. Equivalently, which squares can be represented as the sum of consecutive squares, starting from 1. A centered octahedral number or Haüy octahedral number is a figurate number that counts the number of points of a three-dimensional integer lattice that lie inside an octahedron centered at the origin. The same numbers are special cases of the Delannoy numbers, which count certain two-dimensional lattice paths. The Haüy octahedral numbers are named after René Just Haüy.

In arithmetic and algebra the sixth power of a number n is the result of multiplying six instances of n together. So:

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