|
|
| Line 1: |
Line 1: |
| [[Image:Pyramid of 35 spheres animation.gif|frame|right|A pyramid with side length 5 contains 35 spheres. Each layer represents one of the first five triangular numbers.]]
| | Dalton is what's written across his birth certificate but nevertheless , he never really enjoyed that name. The precious hobby for him and his kids is landscaping but he's been taking on new things a short time ago. Auditing is even his primary income originates from. Massachusetts is where he and his [https://Www.Google.com/search?hl=en&gl=us&tbm=nws&q=wife+survive wife survive]. He's not godd at design but you might want to check their website: http://circuspartypanama.com<br><br>My web page clash of clans hack tool ([http://circuspartypanama.com navigate to this site]) |
| | |
| A '''tetrahedral number''', or '''triangular pyramidal number''', is a [[figurate number]] that represents a [[pyramid (geometry)|pyramid]] with a triangular base and three sides, called a [[tetrahedron]]. The ''n''th tetrahedral number is the sum of the first ''n'' [[triangular number]]s.
| |
| | |
| The first ten tetrahedral numbers are:
| |
| | |
| :[[1 (number)|1]], [[4 (number)|4]], [[10 (number)|10]], [[20 (number)|20]], [[35 (number)|35]], [[56 (number)|56]], [[84 (number)|84]], [[120 (number)|120]], [[165 (number)|165]], [[220 (number)|220]], … {{OEIS|id=A000292}}
| |
| | |
| == Formula ==
| |
| | |
| The formula for the ''n''-th tetrahedral number is represented by the 3rd [[rising factorial]] of ''n'' divided by the [[factorial]] of 3:
| |
| :<math>T_n={n(n+1)(n+2)\over 6} = {n^{\overline 3}\over 3!}</math>
| |
| | |
| The tetrahedral numbers can also be represented as [[binomial coefficient]]s:
| |
| :<math>T_n={n+2\choose3}.</math>
| |
| Tetrahedral numbers can therefore be found in the fourth position either from left or right in [[Pascal's triangle]].
| |
| | |
| == Geometric interpretation ==
| |
| | |
| Tetrahedral numbers can be modelled by stacking spheres. For example, the fifth tetrahedral number (''T''<sub>5</sub> = 35) can be modelled with 35 [[billiard ball]]s 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''<sub>''n''</sub> 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.<ref>http://www.pisquaredoversix.force9.co.uk/Tetrahedra.htm</ref>
| |
| | |
| == Properties == | |
| | |
| *[[A. J. Meyl]] proved in 1878 that only three tetrahedral numbers are also [[Square number|perfect squares]], namely:
| |
| *:''T''<sub>1</sub> = 1² = 1
| |
| *:''T''<sub>2</sub> = 2² = 4
| |
| *:''T''<sub>48</sub> = 140² = 19600.
| |
| | |
| *[[Sir Frederick Pollock, 1st Baronet|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]]:
| |
| *:<math> \!\ \sum_{n=1}^{\infty} \frac{6}{n(n+1)(n+2)} = \frac{3}{2}.</math>
| |
| | |
| * 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 [[parity (mathematics)|parity]] of tetrahedral numbers follows the repeating pattern odd-even-even-even.
| |
| | |
| *An observation of tetrahedral numbers:
| |
| *:''T''<sub>5</sub> = ''T''<sub>4</sub> + ''T''<sub>3</sub> + ''T''<sub>2</sub> + ''T''<sub>1</sub>
| |
| | |
| *Numbers that are both triangular and tetrahedral must satisfy the [[binomial coefficient]] equation:
| |
| *:<math>Tr_n={n+1\choose2}={m+2\choose3}=Te_m.</math>
| |
| | |
| * The only numbers that are both Tetrahedral and Triangular numbers are {{OEIS|id=A027568}}:
| |
| *: ''Te''<sub>1</sub> = ''Tr''<sub>1</sub> = 1
| |
| *: ''Te''<sub>3</sub> = ''Tr''<sub>4</sub> = 10
| |
| *: ''Te''<sub>8</sub> = ''Tr''<sub>15</sub> = 120
| |
| *: ''Te''<sub>20</sub> = ''Tr''<sub>55</sub> = 1540
| |
| *: ''Te''<sub>34</sub> = ''Tr''<sub>119</sub> = 7140
| |
| | |
| ==See also==
| |
| * [[Centered triangular number]]
| |
| | |
| ==References==
| |
| {{Reflist}}
| |
| | |
| ==External links==
| |
| * {{MathWorld |title=Tetrahedral Number |urlname=TetrahedralNumber}}
| |
| * [http://demonstrations.wolfram.com/GeometricProofOfTheTetrahedralNumberFormula/ Geometric Proof of the Tetrahedral Number Formula] by Jim Delany, [[The Wolfram Demonstrations Project]].
| |
| * [http://www.scribd.com/doc/50474167/On-the-Relation-Between-Summations-and-Tetrahedral-Numbers| On the relation between double summations and tetrahedral numbers] by Marco Ripà
| |
| | |
| {{Classes of natural numbers}}
| |
| {{DEFAULTSORT:Tetrahedral Number}}
| |
| [[Category:Figurate numbers]]
| |
Dalton is what's written across his birth certificate but nevertheless , he never really enjoyed that name. The precious hobby for him and his kids is landscaping but he's been taking on new things a short time ago. Auditing is even his primary income originates from. Massachusetts is where he and his wife survive. He's not godd at design but you might want to check their website: http://circuspartypanama.com
My web page clash of clans hack tool (navigate to this site)