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Hometetrahedral number

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# tetrahedral number

An integer of the form

${{(n^{2}+n)(n+2)}\over 6},$ |

where $n$ is a nonnegative integer. Sometimes referred to as $T_{n}$, tetrahedral numbers are listed in A000292 of Sloane’s OEIS. $2|T_{n}$ 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

$T_{n}={n+2\choose 3}.$ |

This means that tetrahedral numbers can be looked up in Pascal’s triangle.

Tetrahedral numbers have practical applications in sphere packing.

Synonym:

triangular pyramidal number

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

11A99*no label found*

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## Comments

## Sloan Sequences.

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.

## Re: SloanE Sequences.

I see PlanetMath and SloanE's OEIS as complementary. The OEIS has the actual numbers, while PlanetMath has the formulas neatly typeset.

## Re: Sloan Sequences.

PlanetMath is not necessarily the place for "deeper understanding." It seems better suited for poseurs like me. I can write a thousand page book on prime numbers in which I use up the entire Greek and Hebrew alphabets for constant or variable names and the only numeric literals are the page numbers. It takes a genius like Neil Sloane to be bold enough to publish a math book filled with numeric literals.