A pyramidal number is the number of points in a

pyramid A pyramid () is a structure whose visible surfaces are triangular in broad outline and converge toward the top, making the appearance roughly a pyramid in the geometric sense. The base of a pyramid can be of any polygon shape, such as trian ...

with a

polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...

al base and triangular sides. The term often refers to

square pyramidal number In mathematics, a pyramid number, or square pyramidal number, is a natural number that counts the stacked spheres in a pyramid (geometry), pyramid with a square base. The study of these numbers goes back to Archimedes and Fibonacci. They are part ...

s, which have a

square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

base with four sides, but it can also refer to a pyramid with any number of sides. The numbers of points in the base and in layers parallel to the base are given by

polygonal number In mathematics, a polygonal number is a Integer, number that counts dots arranged in the shape of a regular polygon. These are one type of 2-dimensional figurate numbers. Polygonal numbers were first studied during the 6th century BC by the Ancien ...

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 (geometry), pyramid with a triangular base and three sides, called a tetrahedron. The th tetrahedral number, , is the sum of the first triangular ...

s) are: : 1, 4, 10, 20, 35, 56, 84, 120, 165,

220 __NOTOC__ Year 220 (Roman numerals, CCXX) was a leap year starting on Saturday 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 ''Ab urbe condita''). The ...

, ... The first few

s are: : 1, 5, 14, 30, 55, 91, 140, 204, 285,

385 Year 385 (Roman numerals, CCCLXXXV) was a common year starting on Wednesday of the Julian calendar. At the time, it was known as the Year of the Consulship of Arcadius, Augustus and Bauto (or, less frequently, year 1138 ''Ab urbe condita''). The ...

, 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 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''). The denomination 196 for this yea ...

288 Year 288 ( CCLXXXVIII) was a leap year starting on Sunday 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 ''Ab urbe condita''). The denomination ...

, 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