Tetrahedral Number (original) (raw)
Last Updated : 23 Jul, 2025
Tetrahedral numbers are a type of figurate number, that represent the number of objects that can be arranged to form a pyramid with a triangular base called a **tetrahedron. Each layer of this pyramid forms a triangle, and the total number of objects increases according to a specific pattern. The nth tetrahedral number is the sum of the first n triangular numbers.
First few tetrahedral numbers are:
1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, 1140, 1330, 1540, 1771, 2024, 2300, 2600, 2925, 3276, 3654, 4060, 4495, 4960, 5456, 5984, 6545, 7140, 7770, 8436, 9139, 9880, 10660, 11480, 12341, 13244, 14190, 15180, . . .
Formula for nth Tetrahedral Number
The **nth tetrahedral number represents the total number of objects in a triangular pyramid with n layers. and it is given by the following formula:
**T n= (n × (n + 1) × (n + 2)) / 6
This formula counts the total number of spheres stacked in a tetrahedron of height n.
**Tetrahedral can be visually represented for various values of Te n :
For **Te 1 **= 1, Te 2 **= 4, Te 3 = 10, Te 4 **= 20
Relationship with Triangular Numbers
Tetrahedral numbers are the sum of the first n triangular numbers. This can be expressed as:
Ten = T1 + T2 + T3 + … + Tn.
Where Tn is the nth triangular number.
The following table displays the first few tetrahedral numbers alongside their corresponding triangular numbers and cumulative sums.
| n | Triangular Number Tn | Tetrahedral Number Ten | Cumulative Sum of Triangular Number |
|---|---|---|---|
| 1 | 1 | 1 | 1 |
| 2 | 3 | 4 | 1 + 3 = 4 |
| 3 | 6 | 10 | 1 + 3 + 6 = 10 |
| 4 | 10 | 20 | 1 + 3 + 6 + 10 = 20 |
| 5 | 15 | 35 | 1 + 3 + 6 + 10 + 15 = 35 |
| 6 | 21 | 56 | 1 + 3 + 6 + 10 + 15 + 21 = 56 |
| 7 | 28 | 84 | 1 + 3 + 6 + 10 + 15 + 21 + 28 = 84 |
| 8 | 36 | 120 | 1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 = 120 |
| 9 | 45 | 165 | 1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 = 165 |
| 10 | 55 | 220 | 1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55 = 220 |
| 11 | 66 | 286 | 1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55 + 66 = 286 |
| 12 | 78 | 364 | 1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55 + 66 + 78 = 364 |
| 13 | 91 | 455 | 1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55 + 66 + 78 + 91 = 455 |
| 14 | 105 | 560 | 1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55 + 66 + 78 + 91 + 105 = 560 |
Interesting Facts about Tetrahedral Numbers
Some interesting facts about Tetrahedral Numbers are:
- Tetrahedral numbers are generally even, except for every fourth tetrahedral number, which is odd (**1, 4, 10, 20, **35, 56, 84, 120, **165 ...).
- The only numbers that are both square and tetrahedral are:
- Te1 = 1.
- Te2 = 4.
- Te48 = 19600.
- Numbers that are both triangular and tetrahedral are:
- Te1 = T1 = 1.
- Te3 = T4 = 10.
- Te8 = T15 = 120.
- Te20 = T55 = 1540.
- Te34 = T119 = 7140.
- Pollock's conjecture states that every every whole number can be written with at most 5 tetrahedral numbers.
- The largest known number that cannot be expressed as the sum of at most four tetrahedral numbers is **343867.
- The only number that is simultaneously both a tetrahedral number and a pyramidal number is **1.
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