A Polygonal Number of the form . The first few are 1, 5, 12, 22, 35, 51, 70, ... (Sloane's A000326).
The Generating Function for the pentagonal numbers is

Every pentagonal number is 1/3 of a Triangular Number.

The so-called generalized pentagonal numbers are given by with , , , ..., the first few of which are 0, 1, 2, 5, 7, 12, 15, 22, 26, 35, ... (Sloane's A001318).

**References**

Guy, R. K. ``Sums of Squares.'' §C20 in
*Unsolved Problems in Number Theory, 2nd ed.* New York: Springer-Verlag, pp. 136-138, 1994.

Pappas, T. ``Triangular, Square & Pentagonal Numbers.'' *The Joy of Mathematics.*
San Carlos, CA: Wide World Publ./Tetra, p. 214, 1989.

Sloane, N. J. A. Sequences
A000326/M3818
and A001318/M1336
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.
*The Encyclopedia of Integer Sequences.* San Diego: Academic Press, 1995.

© 1996-9

1999-05-26