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tetrahedral number (Definition)

An integer of the form

$\displaystyle {{(n^2 + n)(n + 2)} \over 6},$
where $ n$ is a nonnegative integer. Sometimes referred to as $ T_n$, tetrahedral numbers are listed in A000292 of Sloane's OEIS. $ 2\vert 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:

$\displaystyle T_n = \sum_{i = 1}^n t_i.$
Another way to calculate tetrahedral numbers is with the binomial coefficient
$\displaystyle T_n={n+2\choose3}.$
This means that tetrahedral numbers can be looked up in Pascal's triangle.

Tetrahedral numbers have practical applications in sphere packing.



"tetrahedral number" is owned by Mravinci.
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Other names:  triangular pyramidal number
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Cross-references: sphere, applications, Pascal's triangle, binomial coefficient, calculate, triangular number, OEIS, integer
There is 1 reference to this entry.

This is version 4 of tetrahedral number, born on 2006-06-02, modified 2008-08-13.
Object id is 7952, canonical name is TetrahedralNumber.
Accessed 1828 times total.

Classification:
AMS MSC: 11A99 (Number theory :: Elementary number theory :: Miscellaneous)
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Sloan Sequences. by Algeboy on 2006-06-03 12:37:42
I've seen a recent high interest in integer pattern/sequence entries for PM. Most of these quote Sloane's On-line encyclopedia of integer sequences. As I play with the OEIS, I think it is a wonderful site that gives formulas, alternate formulas, related sequences, and loads of article references on just about every sequence you could imagine. I just wonder what value we are are adding in our PM entries?

Is there a way to contect these articles together into some interesting theorems from number theory? Make these entries a more connected part of the theoretical and applied entires on PM?

I fear we could never out do the exposition and depth of the Sloan index if we wish to serve simply as a catalogue. If I were searching for integer sequences on-line I would probably prefer to be directed quickly to Sloans index and by-pass PM, no offense meant to PM and its users, unless PM's articles truly brought out deeper understanding.
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