Apyramidal number is the number of points in apyramid with apolygonal base and triangular sides.^[1] The term often refers tosquare pyramidal numbers, which have asquare base with four sides, but it can also refer to a pyramid with any number of sides.^[2] The numbers of points in the base and in layers parallel to the base are given bypolygonal numbers of the given number of sides, while the numbers of points in each triangular side is given by atriangular number. It is possible to extend the pyramidal numbers to higher dimensions.

The formula for thenthr-gonal pyramidal number is

${\displaystyle P_n^r=\frac{3n^2+n^3(r-2)-n(r-5)}{6},}$

where${\displaystyle r\in \mathbb{N}}$,r ≥ 3.^[1]

This formula can be factored:

${\displaystyle P_n^r=\frac{n(n+1)(n(r-2)-(r-5))}{(2)(3)}=\left(\frac{n(n+1)}{2}\right)\left(\frac{n(r-2)-(r-5)}{3}\right)=T_n\cdot \frac{n(r-2)-(r-5)}{3},}$

whereT_n is thenthtriangular number.

The first few triangular pyramidal numbers (equivalently,tetrahedral numbers) are:

1,4,10,20,35,56,84,120,165,220, ... (sequenceA000292 in theOEIS)

The first fewsquare pyramidal numbers are:

1,5,14,30,55,91,140,204,285,385, 506, 650, 819, ... (sequenceA000330 in theOEIS).

The first few pentagonal pyramidal numbers are:

1,6,18,40,75,126,196,288, 405, 550, 726, 936, 1183, ... (sequenceA002411 in theOEIS).

The first few hexagonal pyramidal numbers are:

1,7,22,50,95,161,252, 372, 525, 715, 946, 1222, 1547, 1925 (sequenceA002412 in theOEIS).

The first few heptagonal pyramidal numbers are:^[3]

1,8,26,60,115, 196, 308, 456, 645, 880, 1166, 1508, 1911, ... (sequenceA002413 in theOEIS)

• Figurate numbers

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