A pentagonal pyramidal number is a figurate number that represents the number of objects in a pyramid with a pentagonal base.^{[1]} The n^{th} 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 n^{th} pentagonal pyramidal number is^{[2]}
so the n^{th} pentagonal pyramidal number is the average of n^{2} and n^{3}.^{[2]} The n^{th} pentagonal pyramidal number is also n times the n^{th} triangular number.
The generating function for the pentagonal pyramidal numbers is^{[1]}
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