Square pyramidal number

# Square pyramidal number

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In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a pyramid number, or square pyramidal number, is a figurate number
Figurate number
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...

that represents the number of stacked spheres in a pyramid
Pyramid
A pyramid is a structure whose outer surfaces are triangular and converge at a single point. The base of a pyramid can be trilateral, quadrilateral, or any polygon shape, meaning that a pyramid has at least three triangular surfaces...

with a square base. Square pyramidal numbers also solve the problem of counting the number of squares in an n × n grid.

## Formula

The first few square pyramidal numbers are:
1, 5, 14
14 (number)
14 is the natural number following 13 and preceding 15.In speech, the numbers 14 and 40 are often confused. When carefully enunciated, they differ in which syllable is stressed: 14 vs 40...

, 30
30 (number)
30 is the natural number following 29 and preceding 31.-In mathematics:30 is the sum of the first four squares, which makes it a square pyramidal number.It is a primorial and is the smallest Giuga number....

, 55
55 (number)
55 is the natural number following 54 and preceding 56.-Albania:*Gazeta 55, a newspaper*Constitution law 55, a law during Communist Albania.-Mathematics:...

, 91
91 (number)
91 is the natural number following 90 and preceding 92.-In mathematics:Ninety-one is the twenty-seventh distinct semiprime and the second of the form...

, 140
140 (number)
140 is the natural number following 139 and preceding 141.-In mathematics:140 is an abundant number and a harmonic divisor number...

, 204
204 (number)
204 is the natural number following 203 and preceding 205. It may be written as "two hundred four" or "two hundred and four".-In mathematics:...

, 285, 385, 506, 650, 819 .

These numbers can be expressed in a formula as
This is a special case of Faulhaber's formula, and may be proved by a straightforward mathematical induction
Mathematical induction
Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers...

. An equivalent formula is given in Fibonacci
Fibonacci
Leonardo Pisano Bigollo also known as Leonardo of Pisa, Leonardo Pisano, Leonardo Bonacci, Leonardo Fibonacci, or, most commonly, simply Fibonacci, was an Italian mathematician, considered by some "the most talented western mathematician of the Middle Ages."Fibonacci is best known to the modern...

's Liber Abaci
Liber Abaci
Liber Abaci is a historic book on arithmetic by Leonardo of Pisa, known later by his nickname Fibonacci...

(1202, ch. II.12).

In modern mathematics, figurate numbers are formalized by the Ehrhart polynomial
Ehrhart polynomial
In mathematics, an integral polytope has an associated Ehrhart polynomial which encodes the relationship between the volume of a polytope and the number of integer points the polytope contains...

s. The Ehrhart polynomial L(P,t) of a polyhedron P is a polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

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)/6 = Pt + 1.

## Relations to other figurate numbers

The square pyramidal numbers can also be expressed as sums of binomial coefficient
Binomial coefficient
In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. They are indexed by two nonnegative integers; the binomial coefficient indexed by n and k is usually written \tbinom nk , and it is the coefficient of the x k term in...

s:
The binomial coefficients occurring in this representation are tetrahedral number
Tetrahedral number
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...

s, 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. In this sum, one of the two tetrahedral numbers counts the number of balls in a stacked pyramid that are directly above or to one side of a diagonal of the base square, and the other tetrahedral number in the sum counts the number of balls that are to the other side of the diagonal. Square pyramidal numbers are also related to tetrahedral numbers in a different way:

The sum of two consecutive square pyramidal numbers is an octahedral number
Octahedral number
In number theory, an octahedral number is a figurate number that represents the number of spheres in an octahedron formed from close-packed spheres...

.

Augmenting a pyramid whose base edge has n balls by adding to one of its triangular faces a tetrahedron
Tetrahedron
In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...

whose base edge has n − 1 balls produces a triangular prism
Triangular prism
In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides....

. Equivalently, a pyramid can be expressed as the result of subtracting a tetrahedron from a prism. This geometric dissection leads to another relation:

Besides 1, there is only one other number that is both a square and a pyramid number: 4900, which is both the 70th square number and the 24th square pyramidal number. This fact was proven by G. N. Watson
G. N. Watson
Neville Watson was an English mathematician, a noted master in the application of complex analysis to the theory of special functions. His collaboration on the 1915 second edition of E. T. Whittaker's A Course of Modern Analysis produced the classic “Whittaker & Watson” text...

in 1918.

In the same way that the square pyramidal numbers can be defined as a sum of consecutive squares, the squared triangular numbers can be defined as a sum of consecutive cubes.

## Squares in a square

A common mathematical puzzle
Mathematical puzzle
Mathematical puzzles make up an integral part of recreational mathematics. They have specific rules as do multiplayer games, but they do not usually involve competition between two or more players. Instead, to solve such a puzzle, the solver must find a solution that satisfies the given conditions....

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 boxes found in the grid is .
• The number of 2×2 boxes found in the grid is . These can be counted by counting all of the possible upper-left corners of 2×2 boxes.
• The number of k×k boxes (1 ≤ kn) found in the grid is . 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 by n square grid is:
That is, the solution to the puzzle is given by the square pyramidal numbers.

The number of rectangles in a square grid is given by the squared triangular numbers.