The pattern found for triangular numbers and for tetrahedral numbers can be generalized. This leads to the formula:
Tetrahedral numbers can be modelled by stacking spheres. For example, the fifth tetrahedral number (Te5 = 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 Ten 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.[dubious – discuss]
Tetrahedral roots and tests for tetrahedral numbers
By analogy with the cube root of x, one can define the (real) tetrahedral root of x as the number n such that Ten = x:
which follows from Cardano's formula. Equivalently, if the real tetrahedral root n of x is an integer, x is the nth tetrahedral number.
The only numbers that are both tetrahedral and triangular numbers are (sequence A027568 in the OEIS):
Te1 = T1 = 1
Te3 = T4 = 10
Te8 = T15 = 120
Te20 = T55 = 1540
Te34 = T119 = 7140
Ten is the sum of all products p × q where (p, q) are ordered pairs and p + q = n + 1
Ten is the number of (n + 2)-bit numbers that contain two runs of 1's in their binary expansion.
Te12 = 364 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". The cumulative total number of gifts after each verse is also Ten for verse n.
The number of possible KeyForge three-house combinations is also a tetrahedral number, Ten−2 where n is the number of houses.