Tetrahedral Number Sequence

The Tetrahedral Number Sequence can be easily understood if you think of a stack of marbles in the shape of a Tetrahedron.

Just count how many marbles would be needed for a stack of a certain height.

  • For height=1 you only need one marble
  • For height=2, you would need 4 marbles (1 at the top and 3 below)
  • For height=3 you would need 10 marbles.
  • For height=4 you would need 20 marbles.
  • How many for height=5 (like the illustration) ... ?

Triangular and Tetrahedral Numbers

Each layer in the tetrahedron of marbles is actually part of the Triangular Number Sequence (1, 3, 6, etc). And both the triangular numbers and the tetrahedral numbers are on Pascal's Triangle.

This table shows the values for the first few layers:

n Triangular Number Tetrahedral Number
(Height) (Marbles in Layer) (Total Marbles)
1 1 1
2 3 4
3 6 10
4 10 20
5 15 35
6 21 56

If you look at the numbers you can see something interesting: if you take any number and add the number below and to the left, you get the next number in the sequence. (For example 6+4=10).


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