Department of Mathematics, Statistics
and Computer Science
Wim Ruitenburg's Fall 2004 MATH025.1001
The Fibonacci Sequence
The Fibonacci sequence is the sequence of whole numbers 0, 1, 1, 2, 3, 5, 8,
13, 21, 34, 55, 89, 144, and so on.
Combinations of the numbers in this sequence occur in nature, see the book on
page 48.
Let us present these Fibonacci numbers as a function F(n).
When we put the function in a table, the result looks like
n
F(n)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
....
0
1
1
2
3
5
8
13
21
34
55
89
144
233
377
610
987
....
So F(3) = 2 and F(11) = 89.
So far we understand the definition of the sequence of Fibonacci numbers
somewhat informally.
An example of a precise mathematical definition is:
F(0) = 0 and F(1) = 1, the initial conditions part
of the definition.
F(n) = F(n-1) + F(n-2), for all n greater than or equal 2
the recursive part of the definition.
With these conditions, the Fibonacci sequence is completely and uniquely
determined.
Mathematicians look for patterns.
Here is one: Each third term is even.
So F(0) = 0 is even, F(3) = 2 is even, F(6) = 8 is even, F(9) = 34 is even,
F(12) = 144 is even, and so on.
These are the only even terms.
Mathematicians often write m = 0 modulo 2 to indicate that m is even.
Similarly, mathematicians often write m = 0 modulo 3 to indicate that m is
divisible by 3.
And so on.
With this notation, the pattern we discovered, can be written as
F(n) = 0 modulo 2, exactly when n = 0 modulo 3
Mathematicians can easily prove this statement.
Once mathematicians see one pattern, they look for more of the same
kind.
Here is one: Each 4th term is divisible by 3.
So F(0), F(4), F(8), F(12), F(16), F(20), and so on, are divisible by 3, and
no other F(n) are divisible by 3.
Each 5th one is divisible by 5.
Each 8th one is divisible by 7.
Here is a list of some of these patterns.
F(n) = 0 modulo 2, exactly when n = 0 modulo 3
F(n) = 0 modulo 3, exactly when n = 0 modulo 4
F(n) = 0 modulo 5, exactly when n = 0 modulo 5
F(n) = 0 modulo 7, exactly when n = 0 modulo 8
F(n) = 0 modulo 11, exactly when n = 0 modulo 10
F(n) = 0 modulo 13, exactly when n = 0 modulo 7
F(n) = 0 modulo 17, exactly when n = 0 modulo 9
F(n) = 0 modulo 19, exactly when n = 0 modulo 18
F(n) = 0 modulo 23, exactly when n = 0 modulo 24
F(n) = 0 modulo 29, exactly when n = 0 modulo 14
Now we have a pattern of patterns.
For each prime number p we find a number k such that
F(n) = 0 modulo p, exactly when n = 0 modulo k
Mathematicians can prove these patterns.
Mathematicians now look for patterns inside this pattern of
patterns.
Here is one:
F(3) = F(2+1) = 0 modulo 2
F(4) = F(3+1) = 0 modulo 3
F(5) = F(5+0) = 0 module 5. An exception in the pattern
F(8) = F(7+1) = 0 modulo 7
F(10) = F(11-1) = 0 modulo 11
F(14) = F(13+1) = 0 modulo 13
F(18) = F(17+1) = 0 modulo 17
F(18) = F(19-1) = 0 modulo 19
F(24) = F(23+1) = 0 modulo 23
F(28) = F(29-1) = 0 modulo 29
So a pattern we see is: For all primes p with the exception of p = 5, we have
that
Either F(p-1) = 0 modulo p, or F(p+1) = 0 modulo p
For example, choose p = 13.
Then either F(12) = 144 is divisible by 13 (it isn't) or F(14) = 377 is
divisible by 13 (it is).
Mathematicians can prove these patterns.
We indicate below ``why'' 5 is such a special exception.
Let us look for a very special G(n) satisfying G(n) = G(n-1) + G(n-2)
for all n greater than or equal 2.
Namely, try G(n) = g^n for some (positive) value of g.
Recall that
g^0 equals 1
g^1 equals g
g^2 equals g times g
g^3 equals g times g times g
and so on.
Whatever g turns out to be, its table will look like
n
G(n)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
....
1
g
g^2
g^3
g^4
g^5
g^6
g^7
g^8
g^9
g^10
g^11
g^12
g^13
g^14
g^15
g^16
....
A number g will make the equation G(n) = G(n-1) + G(n-2) work for all n greater
than or equal to 2, exactly when g^2 = g + 1 or, equivalently
g^2 - g - 1 = 0
When we solve the quadratic equation X^2 - X - 1 = 0, we get two solutions
g_1 = (1 + sqrt(5))/2 as positive root (approximately 1.618033989)
g_2 = (1 - sqrt(5))/2 as negative root (approximately -0.618033989)
The positive root g= g_1 is known as the golden ratio.
Additionally, the second root equals g_2 = -1/g.
Now we have, for example, g^28 = g^27 + g^26 and so on.
But also (-1/g)^54 = (-1/g)^53 + (-1/g)^52, and so on.
Note: The presence of sqrt(5), the square root of 5, in the definition
of the golden ratio g, is related to the prime p = 5 being an exception in the
pattern of pattern of patterns above.
A Fibonacci style sequence is a sequence K(0), K(1), K(2), K(3),
K(4), and so on, which satisfies the recursive equation
K(n) = K(n-1) + K(n-2), for all n greater than or equal to 2.
If, additionally, we pick values for K(0) and K(1), then the whole sequence is
uniquely determined.
Both F(0), F(1), F(2), F(3), and so on, and G(0), G(1), G(2), G(3), and so on,
are examples of Fibonacci style sequences.
Here is an example of another Fibonacci style sequence:
Using the recursive relations, we get K(2) = g^2 + (-1/g)^2; and K(3) = g^3 +
(-1/g)^3; and K(4) = g^4 + (-1/g)^4; and K(5) = g^5 + (-1/g)^5; and so on.
Now -1/g = g_2 is approximately equal to -0.618033989, which is less than 1
from 0.
So higher powers (-1/g)^2, or (-1/g)^3, or (-1/g)^4, must get smaller and
smaller.
Thus if n is big, then K(n) is approximately equal to g^n.
For example, K(5) = 11, while g^5 = 11.090167 approximately.
And K(8) = 47, while g^8 = 46.978714 approximately.
The accuracy gets better with higher powers.
For example, K(14) = 843, while g^14 = 842.9988 approximately.
We started off with the Fibonacci sequence F(0), F(1), F(2), ... near
the top of the page.
We sidetracked and ended up with the numbers g_1 = g (the golden ratio) and g_2
= -1/g.
In good mathematics it frequently occurs that, in the end, one gets back to the
original work.
In this case, we can show that g^0 / sqrt(5) - (-1/g)^0 / sqrt(5) equals 0,
and g^1 / sqrt(5) - (-1/g)^1 / sqrt(5) equals 1.
Let us start a Fibonacci style sequence with
M(0) = g^0 / sqrt(5) - (-1/g)^0 / sqrt(5) = 0
M(1) = g^1 / sqrt(5) - (-1/g)^1 / sqrt(5) = 1
satisfying M(n) = M(n-1) + M(n-2) for all n greater than or equal 2.
With the recursive relations for g_1 and g_2, we get
M(2) = g^2 / sqrt(5) - (-1/g)^2 / sqrt(5)
M(3) = g^3 / sqrt(5) - (-1/g)^3 / sqrt(5)
M(4) = g^4 / sqrt(5) - (-1/g)^4 / sqrt(5)
M(5) = g^5 / sqrt(5) - (-1/g)^5 / sqrt(5)
M(6) = g^6 / sqrt(5) - (-1/g)^6 / sqrt(5)
and so on
Here is a little surprise: Since M(0) = F(0) and M(1) = F(1), the new Fibonacci
style sequence M(0), M(1), M(2), ... and the original Fibonacci sequence F(0),
F(1), F(2), ... must be the same.
So
F(n) = g^n / sqrt(5) - (-1/g)^n / sqrt(5), for all n
Some sample websites on Fibonacci numbers, collected by Phillip van
Hoven: