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Leonardo of Pisa: Bunny Rabbits to Bull Markets - Patterns
What is all the fuss about? Well, look closely at the Fibonacci numbers:
| Terms | f(1) | f(2) | f(3) | f(4) | f(5) | f(6) | f(7) | f(8) | f(9) | f(10) | f(11) | f(12) | f(13) | ||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Values | 1 | 1 | 2 | 3 | 5 | 8 | 13 | 21 | 34 | 55 | 89 | 144 | 233 | ||
Sums of selected Fibonacci numbers gives surprising answers. Add the 1st and 3rd elements (e.g. 1+2 = 3); you get the 4th! That is, in function notation, f(1) + f(3) = f(4). This is no aberration. By adding alternate elements we see a pattern:
| f(1) + f(3) + f(5) = f(6) | 1+2+5=8 |
| f(1) + f(3) + f(5) + f(7) = f(8) | 1+2+5+13=21 |
| f(1) + f(3) + f(5) + f(7) + f(9) = f(10) | 1+2+5+13+34=55 |
| f(1) + f(3) + f(5) + f(7) + f(9) + f(11) = f(12) | 1+2+5+13+34+89=144 |
Adding all successive Fibonacci numbers gives another interesting pattern.
| f(1) + f(2) = f(4) - 1 | 1+1=3-1 |
| f(1) + f(2) + f(3) = f(5) - 1 | 1+1+2=5-1 |
| f(1) + f(2) + f(3) + f(4) = f(6) - 1 | 1+1+2+3=8-1 |
| f(1) + f(2) + f(3) + f(4) + f(5)) = f(7) - 1 | 1+1+2+3+5=13-1 |
Actually, any number can be written as the sum of Fibonacci numbers. For example,
153 = 144 + 8 + 1 which is f(12) + f(6) + f(2).
Eduard Lucas was the first mathematician to study the Fibonacci numbers seriously for their mathematical properties. In fact, he made a general study of number sequences of this kind. They are called recursive sequences; that is, applying a fixed formula to previous terms generates new terms.
Not to be outdone, Lucas defined his own sequence, called, naturally, Lucas numbers. He began his sequence with 1 followed by 3 and then proceeded by adding terms the same way Fibonacci numbers are generated. Check them out below.
| Terms | l(1) | l(2) | l(3) | l(4) | l(5) | l(6) | l(7) | l(8) | l(9) | l(10) | l(11) | l(12) | l(13) | ||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Values | 1 | 3 | 4 | 7 | 11 | 18 | 29 | 47 | 76 | 123 | 199 | 322 | 512 | ||
If you look closely, you can see that Lucas numbers are, in fact, sums of Fibonacci numbers; that is l(2) = f(1) + f(3) (i.e. 3 = 1 + 2); l(3) = f(2) + f(4) (i.e. 4 = 1 + 3); l(4) = f(3) + f(5) (i.e. 7 = 2 + 5). This pattern continues; if we jump all the way to l(12), it is equal to f(11) + f(13) (i.e. 322 = 89 + 233). As you might expect, Lucas numbers and Fibonacci numbers are also linked with interesting connections.
Sandra Monteferrante, "Leonardo of Pisa: Bunny Rabbits to Bull Markets - Patterns," Convergence (August 2011)
