Fibonacci numbers and the Pascal Triangle

 

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Fibonacci Numbers and the Pascal Triangle

        

In the book, Liber abaci ( meaning Book of the Abacus or Book of Calculating ), the practical arithmetic problem is presented : A pair of rabbits is put in a limited area. This pair of rabbits produces another pair each month. If the rabbits do not die, the question is : How many pairs of rabbits there would be ? The answer from the book is this sequence of numbers :
1
1
2
3
5
8
13
21
34
55
 .....
The series of numbers was named " Fibonacci numbers" by Edouard Lucas ( 1842-1899 ).

Lucas invented numerous significant applications of these.The Fibonacci sequence is a recursive sequence where the first two values are 1 and each successive term is obtained by adding together the two previous terms. The definition of the Fibonacci series is :

F(n+1) = F(n-1) + F(n) ,    if n>1 and F(0) = 0, F(1) = 1

By adding diagonal numbers of the Pascal Triangle Fibonacci sequence can be obtained :

     

1
1
1
1
1
1
2
1
2
1
3
3
1
3
1
4
6
4
1
5
1
5
10
10
5
1
8
1
6
15
20
15
6
1
13
1
7
21
35
35
21
7
1
21
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....

     

It can be pressumed where on his journeys Fibonacci met this sequence if "rows" in double Pascal`s Triangle are summed :

     

1
1
1 1
2
1 1 1
3
1 1 2 1
5
1 1 3 2 1
8
1 1 4 3 3 1
13
1 1 5 4 6 3 1
21
.... .... .... .... .... .... .... .... .... .... ....

     

For the sake of playfulness among numbers and areas let us point out the connection between the arithmetical triangle and Fibonacci numbers. The Pascal Triangle is shown as follows and the numbers in rows are summed:

     

1
1
1
1
1
1
2
1
2
3
1
3
1
5
1
4
3
8
1
5
6
1
13
.... .... .... .... .... .... .... .... .... .... ....

     

R. Knott found the Fibonacci numbers appearing as sums of " rows" in Pascal`s Triangle. By drawing Pascal`s Triangle with all the rows moved over by 1 place, we have a clearer arrangement which shows the Fibonacci numbers as sums of columns :

     

  0 1 2 3 4 5 6 7 8 9
0 1                  
1   1 1              
2     1 2 1          
3       1 3 3 1      
4         1 4 6 4 1  
5           1 5 10 10 5
6             1 6 15 20
7               1 7 21
8                 1 8
9                   1
  1 1 2 3 5 8 13 21 34 55

     

Here is the alternative form of Pascal`s Triangle, with the double rows re-aligned as columns and the sums of the new columns are the Fibonacci numbers :

     

  0 1 2 3 4 5 6 7 8 9
0 1                  
1   1                
2   1 1              
3     1 1            
4     1 2 1          
5       1 2 1        
6       1 3 3 1      
7         1 3 3 1    
8         1 4 6 4 1  
9           1 4 6 4 1
  1 2 3 5 8 . . . . .

     

There is a reciprocal connection between Fibonacci numbers and arithmetical triangle. There are also numerous recursive relations for the Fibonacci numbers :

     

F(n+1)
=
1*F(n)
+
1*F(n-1)
F(n+2)
=
1*F(n)
+
2*F(n-1)
+
1*F(n-2)
F(n+3)
=
1*F(n)
+
3*F(n-1)
+
3*F(n-2)
+
1*F(n-3)
F(n+4)
=
1*F(n)
+
4*F(n-1)
+
6*F(n-2)
+
4*F(n-3)
+
1*F(n-4)
F(n+5)
=
1*F(n)
+
5*F(n-1)
+
10*F(n-2)
+
10*F(n-3)
+
5*F(n-4)
+
1*F(n-5)
F(n+6)
=
1*F(n)
+
6*F(n-1)
+
15*F(n-2)
+
20*F(n-3)
+
15*F(n-4)
+
6*F(n-5)
+
1*F(n-6)
.... .... .... .... .... .... .... .... .... .... .... .... .... .... ....

     

It is obvious that the structure of Pascal`s Triangle is built in these recursive relations, which certainly indicates the existing connection between the numbers of Pascal`s Triangle and Fibonacci numbers .

The French mathematician Edouard Lucas found a similar series :

1, 3, 4, 7, 11, 18,29,47 ...

The Fibonacci rule of adding the latest two to get the next is kept, but here we start from 2 and 1 (in this order) instead of 0 and 1 for the (ordinary) Fibonacci numbers :

L(n+1) = L(n-1) + L(n) ,    if n>1 and L(0) = 2, L(1) = 1

Let us point out the connection between the arithmetical triangle and Lucas numbers. The Pascal Triangle is shown as follows and the numbers in rows are summed:

     

1
1
1
1
1
3
1
1
1
1
4
1
1
1
2
1
1
7
1
1
1
3
2
2
1
11
1
1
1
4
3
3
3
1
1
18
1
1
1
5
4
4
6
3
3
1
29
.... .... .... .... .... .... .... .... .... .... ....

     

Also, there is the alternative form of Pascal`s Triangle, with the double rows re-aligned as columns and the sums of the new columns are the Lucas numbers :

     

  0 1 2 3 4 5 6 7 8 9
0     1              
1   1 1              
2       1 1          
3     1 2 1          
4         1 2 1      
5       1 3 3 1      
6           1 3 3 1  
7         1 4 6 4 1  
8             1 4 6 4
9           1 5 10 10 5
  . 1 3 4 7 11 . . . .

     

Here is shewn is a reciprocal connection between Lucas numbers and arithmetical triangle:

     

L(n+1)
=
1*L(n)
+
1*L(n-1)
L(n+2)
=
1*L(n)
+
2*L(n-1)
+
1*L(n-2)
L(n+3)
=
1*L(n)
+
3*L(n-1)
+
3*L(n-2)
+
1*L(n-3)
L(n+4)
=
1*L(n)
+
4*L(n-1)
+
6*L(n-2)
+
4*L(n-3)
+
1*L(n-4)
L(n+5)
=
1*L(n)
+
5*L(n-1)
+
10*L(n-2)
+
10*L(n-3)
+
5*L(n-4)
+
1*L(n-5)
L(n+6)
=
1*L(n)
+
6*L(n-1)
+
15*L(n-2)
+
20*L(n-3)
+
15*L(n-4)
+
6*L(n-5)
+
1*L(n-6)
.... .... .... .... .... .... .... .... .... .... .... .... .... .... ....

     

     

     

        

  2001-2005 Radoslav Jovanovic                 updated:  September 2003.