Tetrahedral number

A pyramid with side length 5 contains 35 spheres. Each layer represents one of the first five triangular numbers.

A tetrahedral number, or triangular pyramidal number, is a figurate number that represents a pyramid with a triangular base and three sides, called a tetrahedron. The nth tetrahedral number is the sum of the first n triangular numbers.

The first ten tetrahedral numbers are:

1, 4, 10, 20, 35, 56, 84, 120, 165, 220, … (sequence A000292 in OEIS)

Formula

The formula for the n-th tetrahedral number is represented by the 3rd rising factorial of n divided by the factorial of 3:

$T_n={n(n+1)(n+2)\over 6} = {n^{\overline 3}\over 3!}$

The tetrahedral numbers can also be represented as binomial coefficients:

$T_n={n+2\choose3}.$

Tetrahedral numbers can therefore be found in the fourth position either from left or right in Pascal's triangle.

Geometric interpretation

Tetrahedral numbers can be modelled by stacking spheres. For example, the fifth tetrahedral number (T₅ = 35) can be modelled with 35 billiard balls and the standard triangular billiards ball frame that holds 15 balls in place. Then 10 more balls are stacked on top of those, then another 6, then another three and one ball at the top completes the tetrahedron.

When order-n tetrahedra built from T_n spheres are used as a unit, it can be shown that a space tiling with such units can achieve a densest sphere packing as long as n ≤ 4.

Properties

• T_n + T_n-1 = 1² + 2² + 3² ... + n²

• A. J. Meyl proved in 1878 that only three tetrahedral numbers are also perfect squares, namely:

  T₁ = 1² = 1

  T₂ = 2² = 4

  T₄₈ = 140² = 19600.

• Sir Frederick Pollock conjectured that every number is the sum of at most 5 tetrahedral numbers: see Pollock tetrahedral numbers conjecture.

• The only tetrahedral number that is also a square pyramidal number is 1 (Beukers, 1988), and the only tetrahedral number that is also a perfect cube is 1.

• The infinite sum of tetrahedral numbers' reciprocals is 3/2, which can be derived using telescoping series:

  $\!\ \sum_{n=1}^{\infty} \frac{6}{n(n+1)(n+2)} = \frac{3}{2}.$

• The tetrahedron with basic length 4 (summing up to 20) can be looked at as the 3-dimensional analogue of the tetractys, the 4th triangular number (summing up to 10).

• The parity of tetrahedral numbers follows the repeating pattern odd-even-even-even.

• An observation of tetrahedral numbers:

  T₅ = T₄ + T₃ + T₂ + T₁

• Numbers that are both triangular and tetrahedral must satisfy the binomial coefficient equation:

  $Tr_n={n+1\choose2}={m+2\choose3}=Te_m.$

• The only numbers that are both Tetrahedral and Triangular numbers are (sequence A027568 in OEIS):

  Te₁ = Tr₁ = 1

  Te₃ = Tr₄ = 10

  Te₈ = Tr₁₅ = 120

  Te₂₀ = Tr₅₅ = 1540

  Te₃₄ = Tr₁₁₉ = 7140

Popular Culture

Te₁₂ = 364, which is the total number of gifts "my true love sent to me" during the course of all 12 verses of the carol, The Twelve Days of Christmas

See also

• Centered triangular number

External links

• Weisstein, Eric W., "Tetrahedral Number", MathWorld.
• Geometric Proof of the Tetrahedral Number Formula by Jim Delany, The Wolfram Demonstrations Project.
• On the relation between double summations and tetrahedral numbers by Marco Ripà