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 * polyg ...

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, quadrilate ...

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 ...

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 bro ...

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 a ...

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 numbers 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 i ...

. 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 numbers) are: : 1, 4, 10, 20, 35, 56, 84,

120 120 may refer to: *120 (number), the number * AD 120, a year in the 2nd century AD *120 BC, a year in the 2nd century BC *120 film, a film format for still photography * ''120'' (film), a 2008 film * 120 (MBTA bus) * 120 (New Jersey bus) * 120 (Ken ...

, 165, 220, ... The first few

s are: : 1, 5, 14, 30, 55, 91, 140,

204 __NOTOC__ Year 204 ( CCIV) was a leap year starting on Sunday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Cilo and Flavius (or, less frequently, year 957 ''Ab urbe c ...

, 285,

385 Year 385 ( CCCLXXXV) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Augustus and Bauto (or, less frequently, year 1138 ''Ab urbe co ...

, 506, 650, 819, ... . The first few pentagonal pyramidal numbers are: : 1, 6, 18, 40, 75, 126, 196, 288, 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,

115 115 may refer to: * 115 (number), the number * AD 115, a year in the 2nd century AD * 115 BC, a year in the 2nd century BC * 115 (Hampshire Fortress) Corps Engineer Regiment, Royal Engineers, a unit in the UK Territorial Army * 115 (Leicestershire) ...

, 196, 308, 456, 645, 880, 1166, 1508, 1911, ...

References

{{Figurate numbers Figurate numbers