Pyramidal number

A pyramidal number is a figurate number that represents a pyramid with a polygonal base and a given number of triangular sides. The term usually refers to square pyramidal numbers, which have a square base with four sides, but it can also refer to:

as well as to pyramids with higher numbers of sides ^[1]

The formula for an r-gonal pyramidal number is:

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

with r ∈ ℕ, r ≥ 3.

This formula can be factorized as follows:

\begin{align}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\ \left[\frac{n(r-2)-(r-5)}{3}\right]\end{align}.

References

This article is issued from Wikipedia - version of the 12/29/2014. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.