Triangular Numbers
Contents of this Page
What are Triangular Numbers?
Special Triangular Numbers
Counting Pairs
Numbers of recangles inside a nxn square
Gauss Sum
Position in Pascal's Triangle
Figurate Numbers
Triangular Numbers on the Internet
To the Main Page    "Mathematische Basteleien"

What are Triangular Numbers?
These are the first 100 triangular numbers:

The sequence of the triangular numbers comes from the natural numbers (and zero), if you always add the next number:


You can illustrate the name triangular number by the following drawing: 

Formulas   top
The general representation of a triangular number is dn= 1 + 2 + 3 + 4 +...+ (n-2) + (n-1) + n,
where n is a natural number.

This sum is dn= n * (n + 1) / 2.

  dn= 1 + 2 + 3 + ...+ (n-2) + (n-1) + n
  dn= n + (n-1) + (n-2) +... + 3 + 2 + 1
Add both sides and combine the right terms in the pairs (n+1). There are n terms. 
2dn=n * (n+1)
dn= n * (n + 1) / 2, q.e.d.
There also is the recursion formula dn+1= dn+ n with d1=1.

Special Triangular Numbers     top
Even and odd triangular numbers
...... You see: 

The even triangular numbers in red and the odd numbers in black form pairs in the usual sequence. 

The smallest square numbers
The smallest palindromic numbers 
d109, d132, d173 , d363, 

Perfect numbers
A number which is equal to the sum of all its divisors smaller than the number itself is called a perfect number. 
The first perfect numbers are 6, 28 and 496. They are triangular numbers like every perfect number. 

The number of 666 
The sum of seven Roman numerals is D+C+L+X+V+I=666. The letter M is missing. 
You also can write: DCLXVI=666.

666 is the largest triangular number which you can form of the same digits (1, page 98). 

666 is a Smith number. This means: The sum of digits [6+6+6] is equal to the sum of the digits of the prime factors [2+3+3+(3+7)]  (1, page 200). 

The number 666 appears in an unfavourable light, because it is called the "number of the animal" in the bible.
Here is wisdom! Who has good brains, should think of the number of the animal; because it is a human's number, and this is 666 (John's revelation 13,18 in Luther's translation)
The number of the animal is a bad number in the interpretations of the bible and is called the "number of the beast",  "Satan's number", or "Antichrist's number".

Consequently people looked in the names of the emperors Nero and Diokletian for 666 and they found it, because they persecuted the Christians. In the 16th century, in the time of the religious wars, 666 was connected with the name of Luther and on the other side with that of the pope. 
The example of the pope uses the idea of the chronogram:  The pope is called VICARIUS FILII DEI (deputy of God). If you add the values of the Roman numerals, you get 666 (VICARIVS FILII DEI).

You are flooded with information on the internet by searching with 666, if you like.

Counting Pairs  top
You give eight squares with the numbers 0,1,2,3,4,5 and 6. They are arranged to pairs. 

There are 7+6+5++4+3+2+1=28 pieces. This is a triangular number. 
There also are dominoes with 36 or 45 pieces, if you add the squares with 7 and 8 numbers. 

Everybody with each other
If you join n points as often as possible, you get 1+2+3+...+(n-1) lines. 

An example with n=7 on the left.

Shaking hands
Everybody shakes hands with each other. Result: You shake hands (1+2+3+...+(n-1)) times.
Everybody clinks glasses of champagne with each other. 

Numbers of recangles inside a nxn square   top
There are 36 rectangles inside a 3x3 square, 14 of them are squared.
Derivation for a n*n square:
Every rectangle is formed by pairs of vertical and horizontal lines.
There are n+1vertikal lines. You can arrange them to n(n+1)/2 pairs. n+1 horizontal lines also have n(n+1)/2 pairs.
There are [n(n+1)/2]² combinations alltogether. If you give n=3, you get 36.
You can easily generalize to the numbers of rectangular solids inside a cube and even inside a rectangular solid. 

Gauss Sum   top
There is a story about the famous mathematician Karl Friedrich Gauß (1777-1855), when he was a child. He should add the numbers 1 to 100. The teacher thought, that he would be busy with it for a long time. But Karl Friedrich found the sum 5050 after some minutes. Instead of adding the numbers one after the other, he made pairs of numbers and could multiply:

  = (1+100) + (2+99) + ... + (50+51)
  = 50*101 
  = 5050.
[(3), page 22f.]

Position in Pascal's Triangle   top
Pascal's triangle makes a contribution to many fields of the number theory.

The red numbers are triangle numbers.

You even can find the sum of the triangular numbers easily.
Example: 1+3+6+10+15=35

You can express the triangular numbers as binomial coefficients 

Figurate Numbers  top
You can generalize the triangular numbers and go further to quadrilateral, pentagons, ...
triangular numbers
square numbers
pentagonal numbers
hexagonal numbers
heptagonal numbers
octogonal numbers

1 3 6 10 15 21 28... 
1 4 9 16 25 36 49... 
1 5 12 22 35 51 70... 
1 6 15 28 45 66 91... 
1 7 18 34 55 81 112... 
1 8 21 40 65 96 133...
It is fun to find out which triangular numbers also appear in the new sequences. 

You can generalize from 2d- (triangle numbers) to higher dimensions:
triangular numbers
tetrahedral numbers
hypertetrahedral numbers
1 3 6 10 15 21...
1 4 10 20 35 56...
1 5 15 35 70 126...
Here you also have the question, which triangular numbers repeat in the new sequences.

There is the famous theorem: 
The sum of two successive numbers is a square number. 
Proof: Add dn and dn+1. The result is (n+1)².  See also the drawings with the triangles above. 

Triangular Numbers on the Internet      top


Jutta Gut
Figurierte Zahlen - die Arithmetik der Spielsteinchen

Dreieckszahl, Zentrierte Dreieckszahl, Polygonalzahl, Sechshundertsechsundsechzig, Hundertdreiundfünfzig, Gaußsche Summenformel


Alexander Bogomolny  (cut-the-knot)
There exist triangular numbers that are also square

Versions of the Gauss Schoolroom Anecdote

Eric W. Weisstein  (MathWorld)
Triangular NumberFigurate Number, Heptagonal Triangular Number, Octagonal Triangular Number
Pentagonal Triangular Number, Pronic Number, Square Triangular Number
Square Triangular Numbers

Patrick De Geest (World of Numbers)
Palindromic Triangulars

Peter Macinnis 
Enquiring into triangular numbers

Shyam Sunder Gupta
Fascinating Triangular Numbers

Triangular number, Centered triangular numberPolygonal number, 153 (number)666 (number)

References   top
(1) Martin Gardner: Die magischen Zahlen des Dr. Matrix, Frankfurt am Main 1987 [ISBN 3-8105-0713-X]
(2) Jan Gullberg: Mathematics - From the Birth of Numbers, New York / London (1997) [ISBN 0-393-04002-X] 
(3) Walter Lietzmann: Lustiges und Merkwürdiges von Zahlen und Formen, Göttingen 1969 

Thank you Gail from Oregon Coast for supporting me in my translation

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©  2004 Jürgen Köller