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Is 120 a tetrahedral number?
What is a tetrahedral number?
Is 120 a triangular number?
What is a tetrahedral number sequence?
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,
Sep 25, 2024 · These numbers correspond to placing discrete points in the configuration of a tetrahedron (triangular base pyramid). Tetrahedral numbers are pyramidal numbers with , and are the sum of consecutive triangular numbers. The first few are 1, 4, 10, 20, 35, 56, 84, 120, ... (OEIS A000292).
This is the Tetrahedral Number Sequence: 1, 4, 10, 20, 35, ... We can understand it better when we think of a stack of marbles in the shape of a Tetrahedron. Just count how many marbles are needed for a stack of a certain height. For height=1 we only need 1 marble.
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A tetrahedral number, or triangular pyramidal number, or Digonal Deltahedral number is a figurate number that represents a pyramid with a triangular base and three sides, called a tetrahedron. The n {\displaystyle n} -th tetrahedral number is the sum of the first n {\displaystyle n} triangular numbers added up.
The first few tetrahedral numbers (sequence A000292 in OEIS) are:
1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, …
The formula for the n {\displaystyle n} -th tetrahedral number is represented by the 3rd Rising Factorial divided by the 3rd Factorial.
T n = n ( n + 1 ) ( n + 2 ) 6 = n 3 ¯ 3 ! {\displaystyle T_{n}={\frac {n(n+1)(n+2)}{6}}={\frac {n^{\overline {3}}}{3!}}}
proof
•Weisstein, Eric W., "Tetrahedral Number" from MathWorld.
•Geometric Proof of the Tetrahedral Number Formula by Jim Delany, The Wolfram Demonstrations Project.
120 is the factorial of 5, i.e.,. the fifteenth triangular number, as well as the sum of the first eight triangular numbers, making it also a tetrahedral number. 120 is the smallest number to appear six times in Pascal's triangle (as all
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, Ten, is the sum of the first n triangular numbers, that is, A pyramid with side length 5 contains 35 spheres.
the fifteenth triangular number, as well as the sum of the first eight triangular numbers, making it also a tetrahedral number. 120 is the smallest number to appear six times in Pascal's triangle (as all triangular and tetragonal numbers appear in it).
