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tetrahedral number (Definition)

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 $$T_n={n+2\choose3}.$$ 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 PrimeFan. [ owner history (1) ]
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Other names:  triangular pyramidal number
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Cross-references: sphere, applications, Pascal's triangle, binomial coefficient, calculate, formula, triangular number, OEIS, integer
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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 2653 times total.

Classification:
AMS MSC11A99 (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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