Tetrahedral Number - Properties

Properties

• 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.

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

• 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:

• 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