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
Read more about this topic: Tetrahedral Number
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—John Locke (16321704)
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—Ralph Waldo Emerson (18031882)
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