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# Pentagonal pyramidal number

A pentagonal pyramidal number is a figurate number that represents the number of objects in a pyramid with a pentagonal base.[1] The nth pentagonal pyramidal number is equal to the sum of the first n pentagonal numbers.

The first few pentagonal pyramidal numbers are:

1, 6, 18, 40, 75, 126, 196, 288, 405, 550, 726, 936, 1183, 1470, 1800, 2176, 2601, 3078, 3610, 4200, 4851, 5566, 6348, 7200, 8125, 9126 (sequence A002411 in the OEIS).

The formula for the nth pentagonal pyramidal number is[2]

${\displaystyle {\frac {n^{2}(n+1)}{2}}}$

so the nth pentagonal pyramidal number is the average of n2 and n3.[2] The nth pentagonal pyramidal number is also n times the nth triangular number.

The generating function for the pentagonal pyramidal numbers is[1]

${\displaystyle {\frac {x(2x+1)}{(x-1)^{4}}}.}$

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