Tetrahedral number

File:Pyramid of 35 spheres animation.gif

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, is a figurate number that represents a pyramid with a triangular base and three sides, called a tetrahedron. The Template:Mvarth tetrahedral number, Template:Mvar, is the sum of the first Template:Mvar triangular numbers, that is,

<math> 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)</math>

The tetrahedral numbers are:

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

FormulaEdit

Template:Pascal triangle simplex numbers.svg

The formula for the Template:Mvarth tetrahedral number is represented by the 3rd rising factorial of Template:Mvar divided by the factorial of 3:

<math>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!}</math>

The tetrahedral numbers can also be represented as binomial coefficients:

<math>Te_n=\binom{n+2}{3}.</math>

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

Proofs of formulaEdit

This proof uses the fact that the Template:Mvarth triangular number is given by

It proceeds by induction.

Base case: <math>Te_1 = 1 = \frac{1\cdot 2\cdot 3}{6}.</math>

Inductive step: <math>\begin{align}

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{align}</math>

The formula can also be proved by Gosper's algorithm.

Recursive relationEdit

Tetrahedral and triangular numbers are related through the recursive formulas

<math>\begin{align}

& Te_n = Te_{n-1} + T_n &(1)\\ & T_n = T_{n-1} + n &(2) \end{align}</math>

The equation <math>(1)</math> becomes

<math>\begin{align}

& Te_n = Te_{n-1} + T_{n-1} + n \end{align}</math>

Substituting <math>n-1</math> for <math>n</math> in equation <math>(1)</math>

<math>\begin{align}

& Te_{n-1} = Te_{n-2} + T_{n-1} \end{align}</math>

Thus, the <math>n</math>th tetrahedral number satisfies the following recursive equation

<math>\begin{align}

& Te_{n} = 2Te_{n-1} - Te_{n-2} + n \end{align}</math>

GeneralizationEdit

The pattern found for triangular numbers <math> \sum_{n_1=1}^{n_2}n_1=\binom{n_2+1}{2}</math> and for tetrahedral numbers <math> \sum_{n_2=1}^{n_3}\sum_{n_1=1}^{n_2}n_1=\binom{n_3+2}{3}</math> can be generalized. This leads to the formula:<ref>Template:Cite journal</ref> <math display=block> \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}</math>

Geometric interpretationEdit

Tetrahedral numbers can be modelled by stacking spheres. For example, the fifth tetrahedral number (Template:Math) 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-Template:Mvar tetrahedra built from Template:Math 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 Template:Math.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= Template:Fix }}

Tetrahedral roots and tests for tetrahedral numbersTemplate:AnchorEdit

By analogy with the cube root of Template:Mvar, one can define the (real) tetrahedral root of Template:Mvar as the number Template:Math such that Template:Math: <math display=block>n = \sqrt[3]{3x+\sqrt{9{x^2}-\frac{1}{27}}} +\sqrt[3]{3x-\sqrt{9{x^2}-\frac{1}{27}}} -1</math>

which follows from Cardano's formula. Equivalently, if the real tetrahedral root Template:Mvar of Template:Mvar is an integer, Template:Mvar is the Template:Mvarth tetrahedral number.

PropertiesEdit

Template:Math, the square pyramidal numbers.

Template:Math, sum of odd squares.

Template:Math, sum of even squares.
A. J. Meyl proved in 1878 that only three tetrahedral numbers are also perfect squares, namely:
Template:Math

Template:Math

Template:Math.
Sir Frederick Pollock conjectured that every positive integer 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 Template:Sfrac, which can be derived using telescoping series:
<math> \sum_{n=1}^{\infty} \frac{6}{n(n+1)(n+2)} = \frac{3}{2}.</math>
The parity of tetrahedral numbers follows the repeating pattern odd-even-even-even.
An observation of tetrahedral numbers:
Template:Math
Numbers that are both triangular and tetrahedral must satisfy the binomial coefficient equation:
<math>T_n=\binom{n+1}{2}=\binom{m+2}{3}=Te_m.</math>

File:Tetrahedral triangular number 10.svg

The third tetrahedral number equals the fourth triangular number as the nth k-simplex number equals the kth n-simplex number due to the symmetry of Pascal's triangle, and its diagonals being simplex numbers; similarly, the fifth tetrahedral number (35) equals the fourth pentatope number, and so forth

The only numbers that are both tetrahedral and triangular numbers are (sequence A027568 in the OEIS):

Template:Math

Template:Math is the sum of all products p × q where (p, q) are ordered pairs and p + q = n + 1
Template:Math 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 <math>2^a+3^b+1</math> for some integers <math>a</math> and <math>b</math> is 8436.

Popular cultureEdit

File:The Twelve Days of Christmas visualisation.svg

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

Template:Math 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".<ref>Template:Cite news</ref> The cumulative total number of gifts after each verse is also Template:Math for verse n.

The number of possible KeyForge three-house combinations is also a tetrahedral number, Template:Math where Template:Mvar is the number of houses.