The sequence of the triangular numbers comes from the natural numbers (and zero), if you always add the next number: 1
1+2= 3
(1+2)+3= 6
(1+2+3)+4= 10
(1+2+3+4)+5= 15
...You can illustrate the name triangular number by the following drawing:
This sum is d _{n}= n * (n
+ 1) / 2.
Proof:
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.
You are flooded with information on the internet by searching
with 666, if you like.
Everybody with each other
Shaking hands Everybody shakes hands with each other. Result: You shake hands (1+2+3+...+(n-1)) times. Prost Everybody clinks glasses of champagne with each other.
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.
= (1+100) + (2+99) + ... + (50+51) = 50*101 = 5050.
You can generalize from 2d- (triangle numbers) to higher dimensions:
There is the famous theorem: The sum of two successive numbers is a square number. Proof: Add d _{n }and d_{n+1}. The result
is (n+1)². See also the drawings with the triangles above.
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