To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.

Tetrahedral number

From Wikipedia, the free encyclopedia

A pyramid with side length 5 contains 35 spheres. Each layer represents one of the first five triangular numbers.
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 n-th tetrahedral number, , is the sum of the first n triangular numbers, that is,

The tetrahedral numbers are:

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

YouTube Encyclopedic

  • 1/5
    1 049
    1 943
    1 325
  • ✪ Geometric Proof of the Tetrahedral Number Formula
  • ✪ 3.4 Q16 Tetrahedral Numbers in Pascal's Triangle
  • ✪ Tetrahedral Numbers
  • ✪ Summation Telescoping Property. Pyramidal and tetrahedral numbers.
  • ✪ Maths of the 12 Days of Christmas




Derivation of Tetrahedral number  from a left-justified Pascal's triangle
Derivation of Tetrahedral number 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:

The tetrahedral numbers can also be represented as binomial coefficients:

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

Proofs of

Proof 1

This proof uses the fact that the n-th triangular number is given by

It proceeds by induction.

Base case
Inductive step

Proof 2

Use Gosper's algorithm

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 Tn 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][dubious ]


  • Tn + Tn-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:
  • 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:
  • 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
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 Ten for verse n.

The number of possible KeyForge three-house-combinations is also a tetrahedral number, Ten-2 where n is the number of houses.

See also


  1. ^ "Tetrahedra". 21 May 2000.
  2. ^ Brent (2006-12-21). "The Twelve Days of Christmas and Tetrahedral Numbers". Retrieved 2017-02-28.

External links

This page was last edited on 13 May 2019, at 09:09
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.