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A pentagonal pyramidal number is a figurate number that represents the number of objects in a pyramid with a pentagonal base.[1] 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 OEIS ).The formula for the n th pentagonal pyramidal number is[2]
n
2
(
n
+
1
)
2
{\displaystyle {\frac {n^{2}(n+1)}{2}}}
so the n th pentagonal pyramidal number is the average of n 2 n 3 [2] n th pentagonal pyramidal number is also n times the n th triangular number .
The generating function for the pentagonal pyramidal numbers is[1]
x
(
2
x
+
1
)
(
x
−
1
)
4
.
{\displaystyle {\frac {x(2x+1)}{(x-1)^{4}}}.}
See also References
Possessing a specific set of other numbers
Expressible via specific sums
