Fibonacci numbers and the Pascal Triangle

 

Google
Search WWW Search milan.milanovic.org

        

Lucas Numbers and the Pascal Triangle

        

Francois-Edouard-Anatole Lucas (4.4.1842 - 8.10.1891) is the French mathematician, professor. Lucas is best known for his results in number theory: in particular he studied the Fibonacci sequence and the associated Lucas sequence is named after him. The main numerical sequence considered by Lucas is the sequence of numbers 1, 3, 4, 7, 11, 18, 29, 47, ... given with the following recurrent formula:

 

L(n) = L(n-1) + L(n-2)

 

for the initial terms L(1) = 1 and L(2) = 3. In the honor of Lucas this numerical sequence was called "Lucas numbers". Note that Lucas numbers have the same significance for mathematics, as well as the classical Fibonacci numbers.

The Lucas numbers are illustrated by the following diagram:


If turned sideways, this may be regarded as the Lucas Tree, which grows according to the rules that

  • initially there is one black node which produces one blue node and two red nodes

  • every red node turns blue after a year

  • every blue node produces one blue node and one red node after a year

At the nth year there are L(n) nodes.

Figure drawn by Henry Bottomley

Pascal`s Triangle is the basic number formula in Nature.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
.... .... .... .... .... .... .... .... .... .... ....

Here is an alternative form:

1
1
1
1
1
3
1
1
2
4
1
1
3
1
1
7
1
1
4
2
3
11
1
1
5
3
6
1
1
18
1
1
6
4
10
3
4
29
1
1
7
5
15
6
10
1
1
47
1
1
8
6
21
10
20
4
5
76
.... .... .... .... .... .... .... .... .... .... ....

Here is Pascal Triangle of the second kind. It is very interesting triangle's form of numbers.The odd numbers , square numbers and pyramidal numbers can be found as columns in Pascal's triangle of the second kind :

     

2
1 2
1 3 2
1 4 5 2
1 5 9 7 2
1 6 14 16 9 2
1 7 20 30 25 11 2
1 8 27 50 55 36 13 2
1 9 35 77 105 91 49 15 2


By adding diagonal numbers in the Pascal Triangle of the Second Kind, Lucas sequence can be obtained :

     

2
2
2
1
1
2
3
1
3
2
5
4
1
4
2
7
9
5
1
7
2
9
16
14
6
1
11
2
11
25
30
20
7
1
18
2
13
36
55
50
27
8
1
29
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....

     

Lucas numbers can be found if "rows" in double Pascal Triangle of the second kind are summed :

     

1
1
1 2
3
1 1 2
4
1 1 3 2
7
1 1 4 3 2
11
1 1 5 4 5 2
18
1 1 6 5 9 5 2
29
.... .... .... .... .... .... .... .... .... .... ....

     

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

     

2
2
1
1
1
2
3
1
3
4
1
4
2
7
1
5
5
11
1
6
9
2
18
.... .... .... .... .... .... .... .... .... .... ....

     

The Lucas numbers appearing as sums of " rows" in Pascal Triangle of the second kind. By drawing Pascal Triangle of the second kind with all the rows moved over by 1 place, we have a clearer arrangement which shows the Lucas numbers as sums of columns :

     

  0 1 2 3 4 5 6 7 8 9
0 2                  
1   1 2              
2     1 3 2          
3       1 4 5 2      
4         1 5 9 7 2  
5           1 6 14 16 9
6             1 7 20 30
7               1 8 27
8                 1 9
9                   1
  2 1 3 4 7 11 18 29 47 76

     

Here is Pascal`s triangle of the second kind alternative form , 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   2                
2   1 2              
3     1 2            
4     1 3 2          
5       1 3 2        
6       1 4 5 2      
7         1 4 5 2    
8         1 5 9 7 2  
9           1 5 9 7 2
  1 3 4 7 11 . . . . .

     

     

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

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: November 2004.