Tetrahedral number

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 nth tetrahedral number is the sum of the first n triangular numbers.

The first ten tetrahedral numbers are:

1, 4, 10, 20, 35, 56, 84, 120, 165, 220, … (sequence A000292 in the OEIS)

Formula

Derivation of tetrahedral numbers from a left-justified Pascal's triangle

The formula for the n-th tetrahedral number is represented by the 3rd rising factorial of n divided by the factorial of 3:

T_{n}={n(n+1)(n+2) \over 6}={n^{\overline {3}} \over 3!}

The tetrahedral numbers can also be represented as binomial coefficients:

T_{n}={n+2 \choose 3}.

Tetrahedral numbers can therefore be found in the fourth position either from left or right in Pascal's triangle.

proof

T_{n}

=\sum _{k=1}^{n}{\frac {k(k+1)}{2}}

=\sum _{k=1}^{n}{\frac {k^{2}+k}{2}}

={\frac {1}{2}}\left(\sum _{k=1}^{n}k^{2}+\sum _{k=1}^{n}k\right)

={\frac {1}{2}}\left({\frac {n(n+1)(2n+1)}{6}}+{\frac {n(n+1)}{2}}\right)

={\frac {1}{2}}\left({\frac {n(n+1)}{2}}\right)\left({\frac {2n+1}{3}}+1\right)

={\frac {n(n+1)}{_{2}{\cancel {4}}}}\left({\frac {{\cancel {2}}n+{\cancel {1+3}}^{2}}{3}}\right)

={\frac {n(n+1)}{2}}\left({\frac {n+2}{3}}\right)

={\frac {n(n+1)(n+2)}{6}}

Geometric interpretation

Tetrahedral numbers can be modelled by stacking spheres. For example, the fifth tetrahedral number (T₅ = 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.

When order-n tetrahedra built from T_n 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 n ≤ 4.^[1]

Properties

T_n + T_n-1 = 1² + 2² + 3² ... + n²
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.
Sir Frederick Pollock conjectured that every number is the sum of at most 5 tetrahedral numbers: see Pollock tetrahedral numbers conjecture.
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:
$\!\ \sum _{n=1}^{\infty }{\frac {6}{n(n+1)(n+2)}}={\frac {3}{2}}.$
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:
$Tr_{n}={n+1 \choose 2}={m+2 \choose 3}=Te_{m}.$
The only numbers that are both Tetrahedral and Triangular numbers are (sequence A027568 in the OEIS):
Te₁ = Tr₁ = 1

Te₃ = Tr₄ = 10

Te₈ = Tr₁₅ = 120

Te₂₀ = Tr₅₅ = 1540

Te₃₄ = Tr₁₁₉ = 7140

Popular culture

Number of gifts of each type and number received each day and their relationship to figurate numbers

Te₁₂ = 364, which is the total number of gifts "my true love sent to me" during the course of all 12 verses of the carol, "The Twelve Days of Christmas".^[2] The cumulative total number of gifts after each verse is also Te_n for verse n.

References

↑
↑ Brent (2006-12-21). "The Twelve Days of Christmas and Tetrahedral Numbers". Mathlesstraveled.com. Retrieved 2017-02-28.

External links

Weisstein, Eric W. "Tetrahedral Number". MathWorld.

Geometric Proof of the Tetrahedral Number Formula by Jim Delany, The Wolfram Demonstrations Project.
On the relation between double summations and tetrahedral numbers by Marco Ripà

Classes of natural numbers

Powers and
related numbers

Of the form
a × 2^b ± 1

Other polynomial
numbers

Recursively defined
numbers

Possessing a
specific set
of other numbers

Expressible via
specific sums

Generated
via a sieve

Lucky

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Meertens

Figurate numbers

2-dimensional

centered	Centered triangular Centered square Centered pentagonal Centered hexagonal Centered heptagonal Centered octagonal Centered nonagonal Centered decagonal Star
non-centered	Triangular Square Square triangular Pentagonal Hexagonal Heptagonal Octagonal Nonagonal Decagonal Dodecagonal

3-dimensional

centered	Centered tetrahedral Centered cube Centered octahedral Centered dodecahedral Centered icosahedral
non-centered	Tetrahedral Octahedral Dodecahedral Icosahedral Stella octangula
pyramidal	Square pyramidal Pentagonal pyramidal Hexagonal pyramidal Heptagonal pyramidal

4-dimensional

centered	Centered pentachoric Squared triangular
non-centered	Pentatope

Pseudoprimes

Combinatorial
numbers

Arithmetic functions

By properties of σ(n)	Abundant Almost perfect Arithmetic Colossally abundant Descartes Hemiperfect Highly abundant Highly composite Hyperperfect Multiply perfect Perfect Practical number Primitive abundant Quasiperfect Refactorable Sublime Superabundant Superior highly composite Superperfect
By properties of Ω(n)	Almost prime Semiprime
By properties of φ(n)	Highly cototient Highly totient Noncototient Nontotient Perfect totient Sparsely totient
By properties of s(n)	Amicable Betrothed Deficient Semiperfect

Dividing a quotient

Other prime factor
or divisor related
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Recreational
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Base-dependent
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