Octahedral Number

A Figurate Number which is the sum of two consecutive Pyramidal Numbers,

$$O_n=P_{n-1}+P_n={\textstyle{1\over 3}} n(2n^2+1).$$

The first few are 1, 6, 19, 44, 85, 146, 231, 344, 489, 670, 891, 1156, ... (Sloane's A005900). The Generating Function for the octahedral numbers is

$${x(x+1)^2\over(x-1)^4}=x+6x^2+19x^3+44x^4+\ldots.$$

See also Truncated Octahedral Number

References

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 50, 1996.

Sloane, N. J. A. Sequence A005900/M4128 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.