A pyramidal number is a

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 (polyhedral numbers). The term can mean * polygon ...

that represents a

pyramid A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilat ...

with a

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...

al 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 -sided polygon of points. The term often refers to

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 Archimedes and Fibonacci. They are part of a broa ...

s, which have a

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...

base with four sides, but it can also refer to pyramids with three or more sides. The numbers of points in the base (and in parallel layers to the base) are given by

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 figurate numbers. Definition and examples T ...

s of the given number of sides, while the numbers of points in each triangular side is given by a

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 th triangular number is the number of dots in ...

. It is possible to extend the pyramidal numbers to higher dimensions.

Formula

The formula for the th -gonal pyramidal number is :

P_n^r= \frac,

where , . This formula can be factored: :

P_n^r=\frac=\left(\frac\right)\left(\frac\right)=T_n \cdot \frac,

where is the th

Sequences

The first few triangular pyramidal numbers (equivalently,

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 th tetrahedral number, , is the sum of the first triangular numbers, that is, ...

s) are: : 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, ... The first few

s are: : 1, 5, 14, 30, 55, 91,

140 140 may refer to: * 140 (number), an integer * AD 140, a year of the Julian calendar * 140 BC, a year of the pre-Julian Roman calendar * ''140'' (video game), a 2013 platform game * Tin King stop Tin King () is an at-grade MTR Light Rail stop ...

, 204,

285 The year 285 ( CCLXXXV) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. In the Roman Empire, it was known as the "Year of the Consulship of Carinus and Aurelius" (or, less frequently, "year ...

, 385, 506, 650, 819, ... . The first few pentagonal pyramidal numbers are: : 1, 6, 18, 40, 75, 126,

196 Year 196 ( CXCVI) was a leap year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Dexter and Messalla (or, less frequently, year 949 ''Ab urbe condita ...

288 Year 288 ( CCLXXXVIII) was a leap year starting on Sunday (link will display the full calendar) of the Julian calendar. In the Roman Empire, it was known as the Year of the Consulship of Maximian and Ianuarianus (or, less frequently, year 1041 ...

, 405, 550, 726, 936, 1183, 1470, 1800, 2176, 2601, 3078, 3610, 4200, 4851, 5566, 6348, 7200, 8125, 9126 . The first few hexagonal pyramidal numbers are: :, , , , , , , 372, 525, 715, 946, 1222, 1547, 1925 . The first few heptagonal pyramidal numbers are:. : 1, 8, 26, 60, 115, 196, 308, 456, 645, 880, 1166, 1508, 1911, ...

References

{{Figurate numbers Figurate numbers