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Tetrahedral number

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Polyhedral number representing a tetrahedron

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

Atetrahedral number, ortriangular pyramidal number, is afigurate number that represents apyramid with a triangular base and three sides, called atetrahedron. Thenth tetrahedral number,Te_n, is the sum of the firstntriangular numbers, that is,

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

The tetrahedral numbers are:

1,4,10,20,35,56,84,120,165,220, ... (sequenceA000292 in theOEIS)

Formula

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Derivation of Tetrahedral number from a left-justifiedPascal's triangle.
  Natural numbers
  Triangular numbers
  **Tetrahedral numbers**
  Pentatope numbers
  5-simplex numbers
  6-simplex numbers
  7-simplex numbers

The formula for thenth tetrahedral number is represented by the 3rdrising factorial ofn divided by thefactorial of 3:

Te_{n}=\sum _{k=1}^{n}T_{k}=\sum _{k=1}^{n}{\frac {k(k+1)}{2}}=\sum _{k=1}^{n}\left(\sum _{i=1}^{k}i\right)={\frac {n(n+1)(n+2)}{6}}={\frac {n^{\overline {3}}}{3!}}

The tetrahedral numbers can also be represented asbinomial coefficients:

Te_{n}={\binom {n+2}{3}}.

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

Proofs of formula

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Six copies of a triangular pyramid withn steps can fit in a cuboid of sizen(n + 1)(n + 2)^[1]

This proof uses the fact that thenth triangular number is given by

T_{n}={\frac {n(n+1)}{2}}.

It proceeds byinduction.

Base case: $Te_{1}=1={\frac {1\cdot 2\cdot 3}{6}}.$

Inductive step: ${\begin{aligned}Te_{n+1}\quad &=Te_{n}+T_{n+1}\\&={\frac {n(n+1)(n+2)}{6}}+{\frac {(n+1)(n+2)}{2}}\\&=(n+1)(n+2)\left({\frac {n}{6}}+{\frac {1}{2}}\right)\\&={\frac {(n+1)(n+2)(n+3)}{6}}.\end{aligned}}$

The formula can also be proved byGosper's algorithm.

Recursive relation

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Tetrahedral and triangular numbers are related through the recursive formulas

{\begin{aligned}&Te_{n}=Te_{n-1}+T_{n}&(1)\\&T_{n}=T_{n-1}+n&(2)\end{aligned}}

The equation $(1) {\displaystyle (1)}$ becomes

{\begin{aligned}&Te_{n}=Te_{n-1}+T_{n-1}+n\end{aligned}}

Substituting $n-1$ for $n {\displaystyle n}$ in equation $(1) {\displaystyle (1)}$

{\begin{aligned}&Te_{n-1}=Te_{n-2}+T_{n-1}\end{aligned}}

Thus, the $n {\displaystyle n}$ th tetrahedral number satisfies the following recursive equation

{\begin{aligned}&Te_{n}=2Te_{n-1}-Te_{n-2}+n\end{aligned}}

Generalization

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The pattern found for triangular numbers $\sum _{n_{1}=1}^{n_{2}}n_{1}={\binom {n_{2}+1}{2}}$ and for tetrahedral numbers $\sum _{n_{2}=1}^{n_{3}}\sum _{n_{1}=1}^{n_{2}}n_{1}={\binom {n_{3}+2}{3}}$ can be generalized. This leads to the formula:^[2] $\sum _{n_{k-1}=1}^{n_{k}}\sum _{n_{k-2}=1}^{n_{k-1}}\ldots \sum _{n_{2}=1}^{n_{3}}\sum _{n_{1}=1}^{n_{2}}n_{1}={\binom {n_{k}+k-1}{k}}$

Geometric interpretation

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Tetrahedral numbers can be modelled by stacking spheres. For example, the fifth tetrahedral number (Te₅ = 35) can be modelled with 35billiard 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 fromTe_n spheres are used as a unit, it can be shown that a space tiling with such units can achieve a densestsphere packing as long asn ≤ 4.^[3]^{[dubious –discuss]}

Tetrahedral roots and tests for tetrahedral numbers

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By analogy with thecube root ofx, one can define the (real) tetrahedral root ofx as the numbern such thatTe_n =x: $n={\sqrt[{3}]{3x+{\sqrt {9{x^{2}}-{\frac {1}{27}}}}}}+{\sqrt[{3}]{3x-{\sqrt {9{x^{2}}-{\frac {1}{27}}}}}}-1$

which follows fromCardano's formula. Equivalently, if the real tetrahedral rootn ofx is an integer,x is thenth tetrahedral number.

Properties

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Te_n +Te_n−1 = 1² + 2² + 3² ... +n², thesquare pyramidal numbers.
Te_2n+1 = 1² + 3² ... +(2n+1)², sum of odd squares.
Te_2n = 2² + 4² ... +(2n)² , sum of even squares.
A. J. Meyl proved in 1878 that only three tetrahedral numbers are alsoperfect squares, namely:
Te₁ = 1² = 1
Te₂ = 2² = 4
Te₄₈ = 140² = 19600.
Sir Frederick Pollock conjectured that every positive integer is the sum of at most 5 tetrahedral numbers: seePollock tetrahedral numbers conjecture.
The only tetrahedral number that is also asquare pyramidal number is 1 (Beukers, 1988), and the only tetrahedral number that is also aperfect cube is 1.
Theinfinite sum of tetrahedral numbers' reciprocals is⁠3/2⁠, which can be derived usingtelescoping series:
$\sum _{n=1}^{\infty }{\frac {6}{n(n+1)(n+2)}}={\frac {3}{2}}.$
Theparity of tetrahedral numbers follows the repeating pattern odd-even-even-even.
An observation of tetrahedral numbers:
Te₅ =Te₄ +Te₃ +Te₂ +Te₁
Numbers that are both triangular and tetrahedral must satisfy thebinomial coefficient equation:
$T_{n}={\binom {n+1}{2}}={\binom {m+2}{3}}=Te_{m}.$

The third tetrahedral number equals the fourth triangular number as thenthk-simplex number equals thekthn-simplex number due to the symmetry ofPascal's triangle, and its diagonals beingsimplex numbers; similarly, the fifth tetrahedral number (35) equals the fourthpentatope number, and so forth

The only numbers that are both tetrahedral and triangular numbers are (sequenceA027568 in theOEIS):

Te₁ =T₁ = 1

Te₃ =T₄ = 10

Te₈ =T₁₅ = 120

Te₂₀ =T₅₅ = 1540

Te₃₄ =T₁₁₉ = 7140

Te_n is the sum of all productsp ×q where (p,q) are ordered pairs andp +q =n + 1
Te_n is the number of (n + 2)-bit numbers that contain two runs of 1's in their binary expansion.
The largest tetrahedral number of the form $2^{a}+3^{b}+1$ for some integers $a {\displaystyle a}$ and $b {\displaystyle b}$ is 8436.

Popular culture

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

Te₁₂ = 364 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".^[4] The cumulative total number of gifts after each verse is alsoTe_n for versen.

The number of possibleKeyForge three-house combinations is also a tetrahedral number,Te_n−2 wheren is the number of houses.

References

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^http://demonstrations.wolfram.com/GeometricProofOfTheTetrahedralNumberFormula
^Baumann, Michael Heinrich (2018-12-12)."Diek-dimensionale Champagnerpyramide"(PDF).Mathematische Semesterberichte (in German).66:89–100.doi:10.1007/s00591-018-00236-x.ISSN 1432-1815.S2CID 125426184.
^"Tetrahedra". 21 May 2000. Archived fromthe original on 2000-05-21.
^Brent (2006-12-21)."The Twelve Days of Christmas and Tetrahedral Numbers".The Math Less Traveled. Archived fromthe original on 2016-11-09. Retrieved2017-02-28.

External links

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Figurate numbers

2-dimensional

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

3-dimensional

centered	Centered tetrahedral numbers Centered cube numbers Centered octahedral numbers Centered dodecahedral numbers Centered icosahedral numbers
non-centered	Cube numbers Octahedral numbers Dodecahedral numbers Icosahedral numbers Stella octangula numbers
pyramidal	Tetrahedral numbers Square pyramidal numbers

4-dimensional

non-centered	Pentatope numbers Squared triangular numbers Tesseractic numbers

Higherdimensional

non-centered	5-hypercube numbers 6-hypercube numbers 7-hypercube numbers 8-hypercube numbers

Classes ofnatural numbers

Powers and related numbers
Achilles Power of 2 Power of 3 Power of 10 Square Cube Fourth power Fifth power Sixth power Seventh power Eighth power Perfect power Powerful Prime power

Of the forma × 2^b ± 1
Cullen Double Mersenne Fermat Mersenne Proth Thabit Woodall

Other polynomial numbers
Hilbert Idoneal Leyland Loeschian Lucky numbers of Euler

Recursively defined numbers
Fibonacci Jacobsthal Leonardo Lucas Narayana Padovan Pell Perrin Graham

Possessing a specific set of other numbers
Amenable Congruent Knödel Riesel Sierpiński

Expressible via specific sums
Nonhypotenuse Polite Practical Primary pseudoperfect Ulam Wolstenholme

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 Cubic Octahedral Dodecahedral Icosahedral Stella octangula
pyramidal	Square pyramidal

4-dimensional

non-centered	Pentatope Squared triangular Tesseractic

Combinatorial numbers
Bell Cake Catalan Dedekind Delannoy Euler Eulerian Fuss–Catalan Lah Lazy caterer's sequence Lobb Motzkin Narayana Ordered Bell Schröder Schröder–Hipparchus Stirling first Stirling second Telephone number Wedderburn–Etherington

Primes
Wieferich Wall–Sun–Sun Wolstenholme prime Wilson

Pseudoprimes
Carmichael number Catalan pseudoprime Elliptic pseudoprime Euler pseudoprime Euler–Jacobi pseudoprime Fermat pseudoprime Frobenius pseudoprime Lucas pseudoprime Lucas–Carmichael number Perrin pseudoprime Somer–Lucas pseudoprime Strong pseudoprime

Arithmetic functions anddynamics

Divisor functions	Abundant Almost perfect Arithmetic Betrothed Colossally abundant Deficient Descartes Hemiperfect Highly abundant Highly composite Hyperperfect Multiply perfect Perfect Practical Primitive abundant Quasiperfect Refactorable Semiperfect Sublime Superabundant Superior highly composite Superperfect
Prime omega functions	Almost prime Semiprime
Euler's totient function	Highly cototient Highly totient Noncototient Nontotient Perfect totient Sparsely totient
Aliquot sequences	Amicable Perfect Sociable Untouchable
Primorial	Euclid Fortunate

Otherprime factor ordivisor related numbers
Blum Cyclic Erdős–Nicolas Erdős–Woods Friendly Giuga Harmonic divisor Jordan–Pólya Lucas–Carmichael Pronic Regular Rough Smooth Sphenic Størmer Super-Poulet

Numeral system-dependent numbers

Arithmetic functions
anddynamics

Persistence
- Additive
- Multiplicative

Digit sum	Digit sum Digital root Self Sum-product
Digit product	Multiplicative digital root Sum-product
Coding-related	Meertens
Other	Dudeney Factorion Kaprekar Kaprekar's constant Keith Lychrel Narcissistic Perfect digit-to-digit invariant Perfect digital invariant Happy