Octagonal Number

$\begin{figure}\begin{center}\BoxedEPSF{OctagonalNumber.epsf scaled 600}\end{center}\end{figure}$

A Polygonal Number of the form . The first few are 1, 8, 21, 40, 65, 96, 133, 176, ... (Sloane's A000567). The Generating Function for the octagonal numbers is

$\begin{displaymath} {x(5x+1)\over(1-x)^3}=x+8x^2+21x^3+40x^4+\ldots. \end{displaymath}$

References

Sloane, N. J. A. Sequence A000567/M4493 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.