Concrete mathematics : a foundation for computer science

The back-to-nature
nature of this example
is shocking.
This book should be
banned.
Phyllotaxis, n.
The love of taxis.
6.6 FIBONACCI NUMBERS 277
Unlike the harmonic numbers and the Bernoulli numbers, the Fibonacci numbers
are nice simple integers. They are defined by the recurrence
F0 = 0;
F, = 1;
F, = F,-I +F,-2, for n > 1. (6.102)
The simplicity of this rule-the simplest possible recurrence in which each
number depends on the previous two-accounts for the fact that Fibonacci
numbers occur in a wide variety of situations.
“Bee trees” provide a good example of how Fibonacci numbers can arise
naturally. Let’s consider the pedigree of a male bee. Each male (also known
as a drone) is produced asexually from a female (also known as a queen); each
female, however, has two parents, a male and a female. Here are the first few
levels of the tree:
The drone has one grandfather and one grandmother; he has one greatgrandfather
and two great-grandmothers; he has two great-great-grandfathers
and three great-great-grandmothers. In general, it is easy to see by induction
that he has exactly Fn+l greatn-grandpas and F,+z greatn-grandmas.
Fibonacci numbers are often found in nature, perhaps for reasons similar
to the bee-tree law. For example, a typical sunflower has a large head that
contains spirals of tightly packed florets, usually with 34 winding in one direction
and 55 in another. Smaller heads will have 21 and 34, or 13 and 21;
a gigantic sunflower with 89 and 144 spirals was once exhibited in England.
Similar patterns are found in some species of pine cones.
And here’s an example of a different nature [219]: Suppose we put two
panes of glass back-to-back. How many ways a,, are there for light rays to
pass through or be reflected after changing direction n times? The first few