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A '''pentagonal pyramidal number''' is a [[figurate number]] that represents the number of objects in a pyramid with a [[pentagonal]] base.<ref name=MathWorld>{{MathWorld|title=Pentagonal Pyramidal Number|urlname=PentagonalPyramidalNumber}}</ref> The ''n''<sup>th</sup> pentagonal pyramidal number is equal to the sum of the first ''n'' [[pentagonal number]]s.


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, 405, 550, 726, 936, 1183, 1470, 1800, 2176, 2601, 3078, 3610, 4200, 4851, 5566, 6348, 7200, 8125, 9126 {{OEIS|id=A002411}}.
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The formula for the ''n''<sup>th</sup> pentagonal pyramidal number is<ref name=OEIS>[[oeis:A002411]]</ref>
 
:<math>\frac{n^2(n+1)}{2}</math>
 
so the ''n''<sup>th</sup> pentagonal pyramidal number is the average of n<sup>2</sup> and n<sup>3</sup>.<ref name=OEIS/> The ''n''<sup>th</sup> pentagonal pyramidal number is also ''n'' times the ''n''<sup>th</sup> [[triangular number]].
 
The [[generating function]] for the pentagonal pyramidal numbers is<ref name=MathWorld/>
 
:<math>\frac{x(2x+1)}{(x-1)^4}.</math>
 
==See also==
*[[Tetrahedral number|Triangular pyramidal number]]
*[[Square pyramidal number]]
 
==References==
<references/>
 
{{Classes of natural numbers}}
[[Category:Figurate numbers]]

Latest revision as of 13:34, 11 April 2014


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