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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,
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]
Each layer represents one of the first five triangular numbers. A truncated triangular pyramid number [1] is found by removing some smaller tetrahedral number (or triangular pyramidal number) from each of the vertices of a bigger tetrahedral number. The number to be removed may be same or different from each of the vertices. [2]
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, ...
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]
In number theory, a pentatope number is a number in the fifth cell of any row of Pascal's triangle starting with the 5-term row 1 4 6 4 1, either from left to right or from right to left. It is named because it represents the number of 3-dimensional unit spheres which can be packed into a pentatope (a 4-dimensional tetrahedron ) of increasing ...
These numbers can be calculated algebraically, as follows. If a pyramid of spheres is decomposed into its square layers with a square number of spheres in each, then the total number of spheres can be counted as the sum of the number of spheres in each square, = = = + + + +, and this summation can be solved to give a cubic polynomial, which can be written in several equivalent ways