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A pyramid with side length 5 contains 35 spheres. Each layer represents one of the first five triangular numbers.

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -th tetrahedral number is the sum of the first Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 -th tetrahedral number is represented by the 3rd Rising Factorial divided by the 3rd Factorial.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_n=\frac{n(n+1)(n+2)}{6}=\frac{n^{\overline3}}{3!}}
proof
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_n} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\sum_{k=1}^n\frac{k(k+1)}{2}=\sum_{k=1}^n\frac{k^2+k}{2}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\frac12\left(\sum_{k=1}^nk^2+\sum_{k=1}^nk\right)=\frac12\left(\frac{n(n+1)(2n+1)}{6}+\frac{n(n+1)}{2}\right)}

Tetrahedral numbers are found in the fourth position either from left to right or right to left in Pascal's triangle. The tetrahedral numbers are therefore binomial coefficients:

Tetrahedral numbers can be modelled by stacking spheres.

For example, the 5th tetrahedral number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_5=35} can be modelled with 35 billiard balls and the standard triangular billiards ball frame that holds 15 balls in place. Then 10 more balls are stacked on top of those, then another 6, then another three and one ball at the top completes the tetrahedron.

A.J. Meyl proved in 1878 that only three tetrahedral numbers are also perfect squares, namely:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_1=1^2=1}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_2=2^2=4}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{48}=140^2=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.

Another interesting fact about tetrahedral numbers is that the infinite sum of their reciprocals is 3/2, which can be derived using telescoping series.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=1}^\infty\frac{6}{n(n+1)(n+2)}=\frac32}

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 tetractys was considered holy by the Pythagoreans.

When order-Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} tetrahedra built from spheres are used as a unit, it can be shown that a space tiling with such units can achieve a densest sphere packing as long as [1].

The parity of tetrahedral numbers follows the repeating pattern odd-even-even-even.

An observation of tetrahedral numbers: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_5=T_4+T_3+T_2+T_1}


Numbers that are both triangular and tetrahedral must satisfy the binomial coefficient equation:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Tr_n=\binom{n+1}{2}=\binom{m+2}{3}=Te_m}

The following are the only numbers that are both Tetrahedral and Triangular numbers:

Tetrahedron1 = Triangle1 = 1

Tetrahedron3 = Triangle4 = 10

Tetrahedron8 = Triangle15 = 120

Tetrahedron20 = Triangle55 = 1540

Tetrahedron34 = Triangle119 = 7140

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