Fibonacci Sequence
________________________________________
0, if n = 0,
F(n) = 1, if n = 1,
F(n-1) + F(n-2), if n > 1
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, ...
Fibonacci Sequence in Pascal's Triangle
Pascal's Triangle is a marvel that develops from a most simple formula:
- start with the number 1 in the first row,
- elements in the following rows are found by adding the two numbers above them to the left and right
There are many patterns that jump out of this formation, but I'll bring you directly to the Fibonacci sequence. It is found by summing elements at an angle.
As shown in the diagram below, the first arrows find a 1 each time. The third arrow passes through 2 ones, making a sum of 2. The fourth arrow passes through a 1 and a 2, making a sum of 3. The next goes though 1, 3 and 1, making a sum of 5. Similarly, the next arrow shows 1+4+3=8, then 1+5+6+1=13 and so on. The sums are shown at the ends of the arrows: 1, 1, 2, 3, 5, 8, 13, ...
What else is hidden in Pascal's triangle? The sum of each row is a power of 2, in ascending order, of course. A line following the left or right side of the triangle finds a sequence of 1s, while the line beside it finds a linear sequence: 1, 2, 3, 4, 5, ... The line beside it finds 1, 3, 6, 10, 16, ... a sequence of order n². Beside it is 1, 4, 10, 20, 35, ... which is order n³ and so on. The numbers in each row are the coefficients which arise in binomial expansions. Another property of the rows is they form a probabilistic distribution.
Copyright © 2007-2009 James Grant
