Definition:Tetrahedral Number

Definition

Tetrahedral numbers are those denumerating a collection of objects which can be arranged in the form of a regular tetrahedron.

The $n$th tetrahedral number $H_n$ is defined as:

$\ds H_n = \sum_{k \mathop = 1}^n T_k$

where $T_k$ is the $k$th triangular number.

Sequence

The sequence of tetrahedral numbers begins as follows:

$0, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, \ldots$

Also see

• Closed Form for Tetrahedral Numbers: $H_n = \dfrac {n \paren {n + 1} \paren {n + 2} } 6$

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $10$
• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $56$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $10$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $56$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): tetrahedral number