Truncated Octahedral Number

A Figurate Number which is constructed as an Octahedral Number with a Square Pyramid removed from each of the six Vertices,

$${\it TO}_n=O_{3n-2}-6P_{n-1}={\textstyle{1\over 3}}(3n-2)[2(3n-2)^2+1],$$

where $O_n$ is an Octahedral Number and $P_n$ is a Pyramidal Number. The first few are 1, 38, 201, 586, ... (Sloane's A005910). The Generating Function for the truncated octahedral numbers is

$${x(6x^3+55x^2+34x+1)\over(x-1)^4}=x+38x^2+201x^3+\ldots.$$

See also Octahedral Number

References

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 52, 1996.

Sloane, N. J. A. Sequence A005910/M5266 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.