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Pyramidal Number

TetrahedralNumber
SquarePyramidalNumber
PentagonalPyramidalNumber
HexagonalPyramidalNumber

A figurate number corresponding to a configuration of points which form a pyramid with r-sided regular polygon bases can be thought of as a generalized pyramidal number, and has the form

P_n^((r))=1/6n(n+1)[(r-2)n+(5-r)].
(1)

The first few cases are therefore

P_n^((3))=1/6n(n+1)(n+2)
(2)
P_n^((4))=1/6n(n+1)(2n+1)
(3)
P_n^((5))=1/2n^2(n+1),
(4)

so r=3 corresponds to a tetrahedral number Te_n, and r=4 to a square pyramidal number P_n.

The pyramidal numbers can also be generalized to four dimensions and higher dimensions (Sloane and Plouffe 1995).

See also

Heptagonal Pyramidal Number, Hexagonal Pyramidal Number, Pentagonal Pyramidal Number, Square Pyramidal Number, Tetrahedral Number

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References

Conway, J. H. and Guy, R. K. "Tetrahedral Numbers" and "Square Pyramidal Numbers" The Book of Numbers. New York: Springer-Verlag, pp. 44-49, 1996.Numberphile. "90,525,801,730 Cannon Balls." Apr. 8, 2019. https://www.youtube.com/watch?v=q6L06pyt9CA.Sloane, N. J. A. and Plouffe, S. "Pyramidal Numbers." Extended entry for sequence M3382 in The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995.

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Pyramidal Number

Cite this as:

Weisstein, Eric W. "Pyramidal Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PyramidalNumber.html

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