Pyramidal number

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 r-sided polygon of points.[1] The term usually refers to square pyramidal numbers, which have a square base with four sides, but it can also refer to pyramids with three or more sides.[2] It is possible to extend the pyramidal numbers to higher dimensions.

Geometric representation of the square pyramidal number 1 + 4 + 9 + 16 = 30.

Formula

The formula for the nth r-gonal pyramidal number is

where r, r ≥ 3. [1]

This formula can be factored:

where Tn is the nth triangular number.

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See also

References

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