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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 n th tetrahedral number, Te n , is the sum of the first n triangular numbers , that is,
Figurate numbers were a concern of the Pythagorean worldview. It was well understood that some numbers could have many figurations, e.g. 36 is a both a square and a triangle and also various rectangles. The modern study of figurate numbers goes back to Pierre de Fermat, specifically the Fermat polygonal number theorem.
In mathematics, a centered tetrahedral number is a centered figurate number that represents a tetrahedron. That is, it counts the dots in a three-dimensional dot pattern with a single dot surrounded by tetrahedral shells. [1] The th centered tetrahedral number, starting at = for a single dot, is: [2] [3]
7.4 Reciprocal of tetrahedral numbers. 7.5 Exponential and logarithms. 8 See also. 9 Notes. 10 References. Toggle the table of contents. ... is a Bernoulli number, ...
Pollock tetrahedral numbers conjecture: Every positive integer is the sum of at most 5 tetrahedral numbers.; The numbers that are not the sum of at most 4 tetrahedral numbers are given by the sequence 17, 27, 33, 52, 73, ..., (sequence A000797 in the OEIS) of 241 terms, with 343,867 conjectured to be the last such number.
If is the th octahedral number and is the th tetrahedral number then O n + 4 T n − 1 = T 2 n − 1 . {\displaystyle O_{n}+4T_{n-1}=T_{2n-1}.} This represents the geometric fact that gluing a tetrahedron onto each of four non-adjacent faces of an octahedron produces a tetrahedron of twice the size.
A triangular-pyramid version of the cannonball problem, which is to yield a perfect square from the N th Tetrahedral number, would have N = 48. That means that the (24 × 2 = ) 48th tetrahedral number equals to (70 2 × 2 2 = 140 2 = ) 19600. This is comparable with the 24th square pyramid having a total of 70 2 cannonballs. [5]
Flu season by the numbers . The CDC estimates that there have been at least 29 million flu infections so far this season, through Feb. 8, including 370,000 hospitalizations and 16,000 deaths.