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| | A '''pyramidal number''' is a [[figurate number]] that represents a [[pyramid]] with a [[Polygonal number|polygonal]] base and a given number of [[Triangular number|triangular]] sides. The term is most often used to refer to [[square pyramidal number]]s, which have a [[square]] base with four sides, but it can also refer to: |
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| | *[[Tetrahedral number|Triangular pyramidal number]] (three sides) |
| | *[[Square pyramidal number]] (four sides) |
| | *[[Pentagonal pyramidal number]] (five sides) |
| | *[[Hexagonal pyramidal number]] (six sides) |
| | *[[Heptagonal pyramidal number]] (seven sides) |
| | |
| | as well as to pyramids with higher numbers of sides <ref>http://oeis.org/A002414</ref> |
| | |
| | The formula for a ''r''-gonal pyramid number is: |
| | |
| | :<math>P_n^r= \frac{3n^2 + n^3(r-2) - n(r-5)}{6}</math> |
| | r ∈ [[ℕ]], r ≥ 3 |
| | |
| | This formula factors to: |
| | |
| | :<math>\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}</math> |
| | |
| | == References == |
| | <references /> |
| | |
| | [[Category:Figurate numbers]] |
![{\begin{aligned}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{aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2873c72890a48e4e5a01cbbc69f8a63f853a805a)