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Pyramidal Number

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={\textstyle{1\over 6}} (n+1)(2p_n^r+n)={\textstyle{1\over 6}}n(n+1)[(r-2)n+(5-r)].
\end{displaymath} (1)

The first few cases are therefore
$\displaystyle P_n^3$ $\textstyle =$ $\displaystyle {\textstyle{1\over 6}}n(n+1)(n+2)$ (2)
$\displaystyle P_n^4$ $\textstyle =$ $\displaystyle {\textstyle{1\over 6}}n(n+1)(2n+1)$ (3)
$\displaystyle P_n^5$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}n^2(n+1),$ (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 4-D and higher dimensions (Sloane and Plouffe 1995).

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


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.

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.

© 1996-9 Eric W. Weisstein