Pentagonal pyramidal number

A pentagonal pyramidal number is a figurate number that represents the number of objects in a pyramid with a pentagonal base. 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 OEIS).

The formula for the n^th pentagonal pyramidal number is

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

so the n^th pentagonal pyramidal number is the average of n² and n³. The n^th pentagonal pyramidal number is also n times the n^th triangular number.

The generating function for the pentagonal pyramidal numbers is

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

See also

• Triangular pyramidal number
• Square pyramidal number