Pentagonal Number

A Polygonal Number of the form $n(3n-1)/2$. The first few are 1, 5, 12, 22, 35, 51, 70, ... (Sloane's A000326). The Generating Function for the pentagonal numbers is

$${x(2x+1)\over(1-x)^3}=x+5x^2+12x^3+22x^4+\ldots.$$

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

The so-called generalized pentagonal numbers are given by $n(3n-1)/2$ with $n=0$, $\pm 1$, $\pm 2$, ..., the first few of which are 0, 1, 2, 5, 7, 12, 15, 22, 26, 35, ... (Sloane's A001318).

See also Euler's Pentagonal Number Theorem, Partition Function P, Polygonal Number, Triangular Number

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.