Fibonacci numbers and the Pascal Triangle

 

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Pascal Triangle

        

The accuracy of the following binomial expressions can be examined by the direct multiplication:

(a+b)0 = 1
(a+b)1 = 1*a + 1*b
(a+b)2 = 1*a2 + 2*ab + 1*b2
(a+b)3 = 1*a3 + 3*a2b + 3*ab2 + 1*b3
(a+b)4 = 1*a4 + 4*a3b + 6*a2b2 + 4*ab3 + 1*b4

If the computational board fields are used for the numbers from the above binomial expressions, the following number triangle is formed :

     

1
1
1
1
2
1
2
1
4
1
3
3
1
8
1
4
6
4
1
16
1
5
10
10
5
1
32
1
6
15
20
15
6
1
64
1
7
21
35
35
21
7
1
128
....
....
....
....
....
....
....
....
....
....
....
....
....

     

There are some proofs that this number triangle was familiar to the Arab astronomer, poet and mathematician Omar Khayyam as early as the XI century. Most probably the number triangle came to Europe from China through Arabia. The Chinese representation of the binomial coefficients, often equally called Pascal`s Triangle being found in his work published for the first time after his death ( in 1665 ) and dealing with figurate numbers, is found for the first time on the title page of the European Arithmetic written by Appianus, in 1527. Late in the XVII century this arithmetical triangle became the key point of the development of the following three mathematical branches : investigation of the infinite series, computation of the finite differences and the theory of probabilities.

Pascal employed his arithmetical triangle in 1653, but no account of his method was printed till 1665. The triangle is constructed as in the figure below, each horizontal line being formed form the one above it by making every number in it equal to the sum of those above and to the left of it in the row immediately above it; ex. gr. the fourth number in the fourth line, namely, 20, is equal to 1 + 3 + 6 + 10.
The numbers in each line are what are now called figurate numbers. Those in the first line are called numbers of the first order; those in the second line, natural numbers or numbers of the second order; those in the third line, numbers of the third order, and so on. It is easily shewn that the mth number in the nth row is (m+n-2)! / (m-1)!(n-1)!

Pascal's arithmetical triangle, to any required order, is got by drawing a diagonal downwards from right to left as in the figure. The numbers in any diagonal give the coefficients of the expansion of a binomial; for example, the figures in the fifth diagonal, namely 1, 4, 6, 4, 1, are the coefficients of the expansion (a + b)^4 Pascal used the triangle partly for this purpose, and partly to find the numbers of combinations of m things taken n at a time, which he stated, correctly, to be (n+1)(n+2)(n+3) ... m / (m-n)!

The first column of the arithmetical triangle is the figure " one" column. Figuratively the figure "one" column represents points. Here is the alternative form of the number triangle where all its columns are the number "one" columns:

     

1
1
1 1 2
1 1 1 3
1 1 1 1 4
1 1 1 1 1 5
1 1 1 1 1 1 6
1 1 1 1 1 1 1 7
.... .... .... .... .... .... .... .... .... .... ....

     

The addition by rows results in the column#2 i.e. the column of the natural numbers : 1,2,3,4,5,6,... Figuratively the series of natural numbers is a straight line. Thus the next triangle is obtained:

     

1
1
2
1
3
3
2
1
6
4
3
2
1
10
5
4
3
2
1
15
6
5
4
3
2
1
21
7
6
5
4
3
2
1
28
....
....
....
....
....
....
....
....
....
....
....

     

Furher addition by rows results in the column#3, the column of the so - called triangular numbers of : 1,3,6,10,15,21,28 ... The triangular numbers symbolize a triangle and an area. The number triangle, whose columns consist of the triangular numbers is shown as follows:

     

1
1
3
1
4
6
3
1
10
10
6
3
1
20
15
10
6
3
1
35
21
15
10
6
3
1
 56
28
21
15
10
6
3
1
84
....
....
....
....
....
....
....
....
....
....
....

     

This time adding by the rows of the upper triangle results in the tetrahedral numbers . These numbers are the column#4 of the Pascal Triangle. The set of tetrahedral numbers symbolize pyramids and dimensionaly the volume : 1,4,10,20,35,56,84 ... The same procedure can be applied to the other columns of the Pascal Triangle, too, whereby a n th dimensional area is formed. The integration is the connection between the neighbouring columns.

An interesting number`s triangle can be generated by adding up natural numbers :

     

1 1 .......
2* 01
2 2 2 .......
2* 03
3 3 3 3 .......
2* 06
4 4 4 4 4 .......
2* 10
5 5 5 5 5 5 ....... 2* 15
6 6 6 6 6 6 6 ....... 2* 21
7 7 7 7 7 7 7 7 ....... 2* 28

Also an interesting number`s triangle can be generated by adding up triangular numbers :

     

01 01 01 .......
3* 01
03 03 03 03 .......
3* 04
06 06 06 06 06 .......
3* 10
10 10 10 10 10 10 .......
3* 20
15 15 15 15 15 15 15 ....... 3* 35
21 21 21 21 21 21 21 21 ....... 3* 56
28 28 28 28 28 28 28 28 28 ....... 3* 84

The number triangle, whose columns consist of the tetrahedral numbers is shown as follows. We get pyramidal numbers:

01 01 01 01 .......
4* 01
04 04 04 04 04 .......
4* 05
10 10 10 10 10 10 .......
4* 15
20 20 20 20 20 20 20 .......
4* 35
35 35 35 35 35 35 35 35 .......
4* 70
56 56 56 56 56 56 56 56 56 .......
4*126

     

        

  2001-2005 Radoslav Jovanovic                 updated:  September 2003.