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A tetrahedral number is a figurate number: a number that can be represented by a regular geometric arrangement of equally spaced points. Tetrahedral numbers correspond to placing discrete points in the configuration of a tetrahedron (triangular base pyramid).

Tetrahedral numbers are the sum of consecutive triangular numbers. The formula is 1/6n(n+1)(n+2). The first few tetrahedral numbers are 1, 4, 10, 20, 35, 56, 84, 120, ...

The tetrahedral numbers are found in the fourth diagonal of Pascal's triangle:

 
 

 

 

 

 

 

 

 

 1

 

 

 

 

 

 

 

 

  Row 0
 

 

 

 

 

 

 

 1

 

 1

 

 

 

 

 

 

 

  Row 1
 

 

 

 

 

 

 1

 

 2

 

 1

 

 

 

 

 

 

  Row 2
 

 

 

 

 

  1

 

 3

 

 3

 

 1

 

 

 

 

 

  Row 3
 

 

 

 

 1

 

 4

 

 6

 

 4

 

 1

 

 

 

 

  Row 4
 

 

 

 1

 

 5

 

10

 

10

 

 5

 

 1

 

 

 

  Row 5
 

 

 1

 

 6

 

15

 

20

 

15

 

 6

 

 1

 

 

  Row 6
 

 1

 

 7

 

21

 

35

 

35

 

21

 

 7

 

 1

 

  Row 7
 1

 

 8

 

28

 

56

 

70

 

56

 

28

 

 8

 

 1

  Row 8

References

  1. Eric's Treasure Trove of Mathematics: Tetrahedral Number
  2. Eric's Treasure Trove of Mathematics: Figurate Number


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Math Forum * * * * 2 May 1998