Jim Delany

Tetrahedral number

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 added up.

The first few tetrahedral numbers are:

1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, …

The formula for the n-th tetrahedral number is

T_n=begin{matrix}{1over6}end{matrix}n(n+1)(n+2).

Tetrahedral numbers are found in the fourth position either from left to right or right to left in Pascal's triangle. The tetrahedral numbers are therefore binomial coefficients:

T_n={n+2choose3}

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.

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.
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.

Another interesting fact about tetrahedral numbers is that the infinite sum of their reciprocals is 3/2, which can be derived using telescoping series.

! sum_{n=1}^{infty}{6 over {n(n+1)(n+2)}} = {3 over 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 tetractys was considered holy by the Pythagoreans.

When order-n tetrahedra built from Tn 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

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+1choose2}={m+2choose3}=Te_m

The following are the only numbers that are both Tetrahedral and Triangular numbers:

Tetrahedron1 = Triangle1 = 1

Tetrahedron3 = Triangle4 = 10

Tetrahedron8 = Triangle15 = 120

Tetrahedron20 = Triangle55 = 1540

Tetrahedron34 = Triangle119 = 7140

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