# Category: Figurate numbers

Dodecagonal number
A dodecagonal number is a figurate number that represents a dodecagon. The dodecagonal number for n is given by the formula The first few dodecagonal numbers are: 0, 1, 12, 33, 64, 105, 156, 217, 288,
Triangular number
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 n
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. The nth octahedral number can be obtained by the fo
Seventh power
In arithmetic and algebra the seventh power of a number n is the result of multiplying seven instances of n together. So: n7 = n × n × n × n × n × n × n. Seventh powers are also formed by multiplying
Centered triangular number
A centered (or centred) triangular number is a centered figurate number that represents an equilateral triangle with a dot in the center and all its other dots surrounding the center in successive equ
Beer can pyramid
A beer can pyramid, often called a beeramid as a portmanteau, is a pyramid made from discarded beer cans. Beer can pyramids are built as empty beer cans became available, slowly growing as the night (
Pentatope number
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. The first few numbers of this
Pronic number
A pronic number is a number that is the product of two consecutive integers, that is, a number of the form The study of these numbers dates back to Aristotle. They are also called oblong numbers, hete
Centered dodecahedral number
A centered dodecahedral number is a centered figurate number that represents a dodecahedron. The centered dodecahedral number for a specific n is given by The first such numbers are 1, 33, 155, 427, 9
Pentagonal number
A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal n
Centered icosahedral number
The centered icosahedral numbers and cuboctahedral numbers are two different names for the same sequence of numbers, describing two different representations for these numbers as three-dimensional fig
Sixth power
In arithmetic and algebra the sixth power of a number n is the result of multiplying six instances of n together. So: n6 = n × n × n × n × n × n. Sixth powers can be formed by multiplying a number by
Gnomon (figure)
In geometry, a gnomon is a plane figure formed by removing a similar parallelogram from a corner of a larger parallelogram; or, more generally, a figure that, added to a given figure, makes a larger f
Star number
A star number is a centered figurate number, a centered hexagram (six-pointed star), such as the Star of David, or the board Chinese checkers is played on. The nth star number is given by the formula
Nonagonal number
A nonagonal number (or an enneagonal number) is a figurate number that extends the concept of triangular and square numbers to the nonagon (a nine-sided polygon). However, unlike the triangular and sq
Centered octagonal number
A centered octagonal number is a centered figurate number that represents an octagon with a dot in the center and all other dots surrounding the center dot in successive octagonal layers. The centered
Dodecahedral number
A dodecahedral number is a figurate number that represents a dodecahedron. The nth dodecahedral number is given by the formula The first such numbers are 0, 1, 20, 84, 220, 455, 816, 1330, 2024, 2925,
Fermat polygonal number theorem
In additive number theory, the Fermat polygonal number theorem states that every positive integer is a sum of at most n n-gonal numbers. That is, every positive integer can be written as the sum of th
Stella octangula number
In mathematics, a stella octangula number is a figurate number based on the stella octangula, of the form n(2n2 − 1). The sequence of stella octangula numbers is 0, 1, 14, 51, 124, 245, 426, 679, 1016
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. The nth tetrahedral number, Ten, is t
Centered octahedral number
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
Cube (algebra)
In arithmetic and algebra, the cube of a number n is its third power, that is, the result of multiplying three instances of n together.The cube of a number or any other mathematical expression is deno
Heptagonal number
A heptagonal number is a figurate number that is constructed by combining heptagons with ascending size. The n-th heptagonal number is given by the formula . The first few heptagonal numbers are: 0, 1
Eighth power
In arithmetic and algebra the eighth power of a number n is the result of multiplying eight instances of n together. So: n8 = n × n × n × n × n × n × n × n. Eighth powers are also formed by multiplyin
Centered polyhedral number
The centered polyhedral numbers are a class of figurate numbers, each formed by a central dot, surrounded by polyhedral layers with a constant number of edges. The length of the edges increases by one
Polite number
In number theory, a polite number is a positive integer that can be written as the sum of two or more consecutive positive integers. A positive integer which is not polite is called impolite. The impo
Fourth power
In arithmetic and algebra, the fourth power of a number n is the result of multiplying four instances of n together. So: n4 = n × n × n × n Fourth powers are also formed by multiplying a number by its
Square triangular number
In mathematics, a square triangular number (or triangular square number) is a number which is both a triangular number and a perfect square. There are infinitely many square triangular numbers; the fi
Centered polygonal number
The centered polygonal numbers are a class of series of figurate numbers, each formed by a central dot, surrounded by polygonal layers of dots with a constant number of sides. Each side of a polygonal
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 (polygonal numbers) and different dimensions (p
Centered square number
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
Centered heptagonal number
A centered heptagonal number is a centered figurate number that represents a heptagon with a dot in the center and all other dots surrounding the center dot in successive heptagonal layers. The center
Cannonball problem
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 wha
Centered cube number
A centered cube number is a centered figurate number that counts the number of points in a three-dimensional pattern formed by a point surrounded by concentric cubical layers of points, with i2 points
Centered nonagonal number
A centered nonagonal number (or centered enneagonal number) is a centered figurate number that represents a nonagon with a dot in the center and all other dots surrounding the center dot in successive
Centered decagonal number
A centered decagonal number is a centered figurate number that represents a decagon with a dot in the center and all other dots surrounding the center dot in successive decagonal layers. The centered
Centered tetrahedral number
A centered tetrahedral number is a centered figurate number that represents a tetrahedron. The centered tetrahedral number for a specific n is given by The first such numbers are 1, 5, 15, 35, 69, 121
Descartes on Polyhedra
Descartes on Polyhedra: A Study of the "De solidorum elementis" is a book in the history of mathematics, concerning the work of René Descartes on polyhedra. Central to the book is the disputed priorit
Hexagonal number
A hexagonal number is a figurate number. The nth hexagonal number hn is the number of distinct dots in a pattern of dots consisting of the outlines of regular hexagons with sides up to n dots, when th
Polygonal number
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
Centered pentagonal number
A centered pentagonal number is a centered figurate number that represents a pentagon with a dot in the center and all other dots surrounding the center in successive pentagonal layers. The centered p
Pollock's conjectures
Pollock's conjectures are two closely related unproven conjectures in additive number theory. They were first stated in 1850 by Sir Frederick Pollock, better known as a lawyer and politician, but also
Square pyramidal number
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 Arc
Icosahedral number
An icosahedral number is a figurate number that represents an icosahedron. The nth icosahedral number is given by the formula The first such numbers are 1, 12, 48, 124, 255, 456, 742, 1128, 1629, 2260
Decagonal number
A decagonal number is a figurate number that extends the concept of triangular and square numbers to the decagon (a ten-sided polygon). However, unlike the triangular and square numbers, the patterns
Square number
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, sinc
Octagonal number
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 1, 8, 21, 40, 65, 96,
Centered hexagonal number
In mathematics and combinatorics, a centered hexagonal number, or hex number, is a centered figurate number that represents a hexagon with a dot in the center and all other dots surrounding the center
Fifth power (algebra)
In arithmetic and algebra, the fifth power or sursolid of a number n is the result of multiplying five instances of n together: n5 = n × n × n × n × n. Fifth powers are also formed by multiplying a nu
Ehrhart polynomial
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 the
Pyramidal number
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