View source for Pyramidal number

{{Short description|Figurate number}}
[[File:Square pyramidal number.svg|thumb|upright=1.35|Geometric representation of the square pyramidal number {{nowrap|1=1 + 4 + 9 + 16 = 30.}}]]A '''pyramidal number''' is the number of points in a [[pyramid_(geometry)|pyramid]] with a [[polygon]]al base and triangular sides.<ref name=":0" /> The term often refers to [[square pyramidal number]]s, which have a [[square]] base with four sides, but it can also refer to a pyramid with any number of sides.<ref>{{Cite OEIS|1=A002414}}</ref> The numbers of points in the base and in layers parallel to the base are given by [[polygonal number]]s of the given number of sides, while the numbers of points in each triangular side is given by a [[triangular number]]. It is possible to extend the pyramidal numbers to higher dimensions.

== Formula ==

The formula for the {{mvar|n}}th {{mvar|r}}-gonal pyramidal number is

:<math>P_n^r= \frac{3n^2 + n^3(r-2) - n(r-5)}{6},</math>
where {{math|''r'' ∈ <math>\mathbb{N}</math>}}, {{math|''r'' ≥ 3}}. <ref name=":0">{{MathWorld |id=PyramidalNumber |title=Pyramidal Number}}</ref>

This formula can be factored:

:<math>P_n^r=\frac{n(n+1)\bigl(n(r-2)-(r-5)\bigr)}{(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},</math>

where {{mvar|T<sub>n</sub>}} is the {{mvar|n}}th [[triangular number]].

==Sequences==
The first few triangular pyramidal numbers (equivalently, [[tetrahedral number]]s) are:

:[[1]], [[4]], [[10]], [[20 (number)|20]], [[35 (number)|35]], [[56 (number)|56]], [[84 (number)|84]], [[120 (number)|120]], [[165 (number)|165]], [[220 (number)|220]], ... {{OEIS|id=A000292}}

The first few [[square pyramidal number]]s are:
:[[1 (number)|1]], [[5 (number)|5]], [[14 (number)|14]], [[30 (number)|30]], [[55 (number)|55]], [[91 (number)|91]], [[140 (number)|140]], [[204 (number)|204]], [[280 (number)#285|285]], [[300 (number)#385|385]], 506, 650, 819, ... {{OEIS|id=A000330}}.

The first few pentagonal pyramidal numbers are:

:[[1 (number)|1]], [[6 (number)|6]], [[18 (number)|18]], [[40 (number)|40]], [[75 (number)|75]], [[126 (number)|126]], [[196 (number)|196]], [[288 (number)|288]], 405, 550, 726, 936, 1183, 1470, 1800, 2176, 2601, 3078, 3610, 4200, 4851, 5566, 6348, 7200, 8125, 9126 {{OEIS|id=A002411}}.

The first few hexagonal pyramidal numbers are:
:{{num|1}}, {{num|7}}, {{num|22}}, {{num|50}}, {{num|95}}, {{num|161}}, {{num|252}}, 372, 525, 715, 946, 1222, 1547, 1925 {{OEIS|A002412}}.

The first few heptagonal pyramidal numbers are:<ref name="b">{{citation|title=Recreations in the Theory of Numbers: The Queen of Mathematics Entertains|first=Albert H.|last=Beiler|publisher=Courier Dover Publications|year=1966|isbn=9780486210964|page=194|url=https://books.google.com/books?id=fJTifbYNOzUC&pg=PA194}}.</ref>
:[[1 (number)|1]], [[8 (number)|8]], [[26 (number)|26]], [[60 (number)|60]], [[115 (number)|115]], 196, 308, 456, 645, 880, 1166, 1508, 1911, ...  {{OEIS|id=A002413}}

== References ==
{{reflist}}

{{Figurate numbers}}
[[Category:Figurate numbers]]