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 $n$ 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]

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

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

References

^ ^a ^b Weisstein, Eric W. "Pentagonal Pyramidal Number". MathWorld.
^ ^a ^b oeis:A002411

[MathWorld-1] Weisstein, Eric W. "Pentagonal Pyramidal Number". MathWorld.

[OEIS-2] s:A002411

[1]

[2]

See also

References