Tetrahedral Number

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:

Tn = (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 Ten:

For Te1 = 1, Te2 = 4, Te3 = 10, Te4 = 20

tetrahedral-number

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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