Pyramidal number
Pyramidal number
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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:
Triangular pyramidal number (three 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 [1]
The formula for an r-gonal pyramidal number is:
- Pnr=3n2+n3(r−2)−n(r−5)6,{displaystyle 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:
- Pnr=n(n+1)[n(r−2)−(r−5)](2)(3)=[n(n+1)2][n(r−2)−(r−5)3]=Tn [n(r−2)−(r−5)3].{displaystyle {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}}.}
![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}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/6dcffe357afbd0b95cde7cf556b62f859f67dc87)
References[edit]
^ Sloane, N. J. A. (ed.). "Sequence A002414". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
Categories:
- Figurate numbers
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