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 -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 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. 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 triangular number.

Sequences

The first few triangular pyramidal numbers (equivalently, tetrahedral numbers) are: : 1, 4, 10, 20, 35, 56, 84, 120,

165 Year 165 ( CLXV) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Orfitus and Pudens (or, less frequently, year 918 ''Ab urbe condita'' ...

220 __NOTOC__ Year 220 ( CCXX) was a leap year starting on Saturday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Antonius and Eutychianus (or, less frequently, year 973 '' ...

, ... 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 __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 co ...

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 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,

126 126 may refer to: *126 (number), a natural number *AD 126, a year in the 2nd century AD *126 BC, a year in the 2nd century BC *126 film, a cartridge-based film format used in still photography * 126 (New Jersey bus) * 126 Artist-run Gallery *Interst ...

, 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) Field ...

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

References

{{Figurate numbers Figurate numbers