A pentagonal pyramidal number is a figurate number that represents the number of objects in a pyramid with a pentagonal base. 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 OEIS).
The formula for the nth pentagonal pyramidal number is
so the nth pentagonal pyramidal number is the average of n2 and n3. The nth pentagonal pyramidal number is also n times the nth triangular number.
The generating function for the pentagonal pyramidal numbers is
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