The first few tetrahedral numbers are:
The formula for the n-th tetrahedral number is
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.
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:
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