TetrahedralNumber 2006-06-02 16:39:35 2008-08-13 19:28:22 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here An integer of the form $${{(n^2 + n)(n + 2)} \over 6},$$ where $n$ is a nonnegative integer. Sometimes referred to as $T_n$, {\em tetrahedral numbers} are listed in A000292 of Sloane's OEIS. $2|T_n$ except when $n \equiv 1 \mod 4$. With $t_n$ the $n$th triangular number, the $n$th tetrahedral number can be calculated with this formula: $$T_n = \sum_{i = 1}^n t_i.$$ Another way to calculate tetrahedral numbers is with the binomial coefficient $$T_n={n+2\choose3}.$$ This means that tetrahedral numbers can be looked up in Pascal's triangle. Tetrahedral numbers have practical applications in sphere packing.