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

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

The first few cases are therefore

$$P_n^3 = {\textstyle{1\over 6}}n(n+1)(n+2)$$ (2)

$$P_n^4 = {\textstyle{1\over 6}}n(n+1)(2n+1)$$ (3)

$$P_n^5 = {\textstyle{1\over 2}}n^2(n+1),$$ (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

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.

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.