Rhombic Dodecahedral Number

A Figurate Number which is constructed as a centered Cube with a Square Pyramid appended to each face,

$${\it RhoDod}_n={\it CCub}_n+6P_{n-1}=(2n-1)(2n^2-2n+1),$$

where ${\it CCub}_n$ is a Centered Cube Number and $P_n$ is a Pyramidal Number. The first few are 1, 15, 65, 175, 369, 671, ... (Sloane's A005917). The Generating Function of the rhombic dodecahedral numbers is

$${x(1+11x+11x^2+x^3)\over(x-1)^4}=x+15x^2+65x^3+175x^4+\ldots.$$

See also Octahedral Number

References

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 53-54, 1996.

Sloane, N. J. A. Sequence A005917/M4968 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.