A pentagonal pyramidal number counts objects stacked in a pyramid with [[pentagonal]] layers — 1 on top, then 5, then 12, and so on — given by \(k^2(k+1)/2\). The sequence: 1, 6, 18, 40, 75, 126, 196, 288, 405.
It's the pentagonal member of the pyramidal-number family that also includes the [[tetrahedral]] (triangular-based) and [[square-pyramidal]] numbers. Each pentagonal pyramidal number is the running total of the first \(k\) pentagonal numbers.