Truncated Tetrahedral Number

A Figurate Number constructed by taking the $(3n-2)$th Tetrahedral Number and removing the $(n-1)$th Tetrahedral Number from each of the four corners,

$${\rm Ttet}_n\equiv {\rm Te}_{3n-3}-4{\rm Te}_{n-1}={\textstyle{1\over 6}} n(23n^2-27n+10).$$

The first few are 1, 16, 68, 180, 375, ... (Sloane's A005906). The Generating Function for the truncated tetrahedral numbers is

$${x(10x^2+12x+1)\over(x-1)^4}=x+16x^2+68x^3+180x^4+\ldots.$$

References

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 46-47, 1996.

Sloane, N. J. A. Sequence A005906/M5002 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.